1
00:00:00,499 --> 00:00:02,830
The following content is
provided under a Creative

2
00:00:02,830 --> 00:00:04,340
Commons license.

3
00:00:04,340 --> 00:00:06,680
Your support will help
MIT OpenCourseWare

4
00:00:06,680 --> 00:00:11,050
continue to offer high quality
educational resources for free.

5
00:00:11,050 --> 00:00:13,660
To make a donation, or
view additional materials

6
00:00:13,660 --> 00:00:17,556
from hundreds of MIT courses,
visit MIT OpenCourseWare

7
00:00:17,556 --> 00:00:18,181
at ocw.mit.edu.

8
00:00:24,160 --> 00:00:26,400
PROFESSOR: So far in our
discussion of probability,

9
00:00:26,400 --> 00:00:29,210
we've focused our attention
on the probability

10
00:00:29,210 --> 00:00:33,220
that some kind of event
occurs-- the probability you

11
00:00:33,220 --> 00:00:36,600
win the Monty Hall game,
the probability that you get

12
00:00:36,600 --> 00:00:40,870
a heads when you flip a
coin, the probability we all

13
00:00:40,870 --> 00:00:44,570
have different birthdays in
the room-- that kind of thing.

14
00:00:44,570 --> 00:00:47,760
In each case, the event
happens or it doesn't.

15
00:00:47,760 --> 00:00:50,630
It's a zero-one kind of thing.

16
00:00:50,630 --> 00:00:52,710
For the rest of
the course, we're

17
00:00:52,710 --> 00:00:56,460
going to start talking about
more complicated situations.

18
00:00:56,460 --> 00:00:58,840
Instead of talking
about zero-one events,

19
00:00:58,840 --> 00:01:00,890
like winning or
losing, we're going

20
00:01:00,890 --> 00:01:04,610
to start talking about
how much you win.

21
00:01:04,610 --> 00:01:06,926
Instead of talking
about whether or not

22
00:01:06,926 --> 00:01:08,970
you get a heads in
a coin flip, we're

23
00:01:08,970 --> 00:01:10,930
going to talk about
flipping lots of coins

24
00:01:10,930 --> 00:01:16,080
and asking how many
heads did you get?

25
00:01:16,080 --> 00:01:21,420
Now, this involves the
notion of a random variable

26
00:01:21,420 --> 00:01:24,990
to count how much or how
many in a random situation.

27
00:01:24,990 --> 00:01:27,250
And actually, the
name is sort of weird.

28
00:01:27,250 --> 00:01:29,650
It's not a variable at all.

29
00:01:29,650 --> 00:01:33,080
In fact, a random variable
is a function from the sample

30
00:01:33,080 --> 00:01:34,190
space to the real numbers.

31
00:01:45,390 --> 00:01:56,220
See random variable R is a
function R from the sample

32
00:01:56,220 --> 00:02:00,512
space to the reals.

33
00:02:00,512 --> 00:02:02,470
So this is the random
variable, and we'll often

34
00:02:02,470 --> 00:02:05,040
denote that by rv.

35
00:02:05,040 --> 00:02:07,980
This is the sample space
of all possible outcomes.

36
00:02:11,190 --> 00:02:12,460
And this is just the reals.

37
00:02:17,460 --> 00:02:19,600
All right, so in other
words, a random variable

38
00:02:19,600 --> 00:02:22,150
is just a way of
assigning a real number

39
00:02:22,150 --> 00:02:25,440
to each possible outcome.

40
00:02:25,440 --> 00:02:28,085
For example, say that
we toss three coins.

41
00:02:36,560 --> 00:02:45,080
We could let R equal the number
of heads among the three coins.

42
00:02:45,080 --> 00:02:47,040
And R is thus a random variable.

43
00:02:47,040 --> 00:02:51,920
For example, for the outcome
where the first coin is heads,

44
00:02:51,920 --> 00:02:54,050
the second is tails,
and the third is heads,

45
00:02:54,050 --> 00:02:57,610
R would be the real
number 2, indicating I

46
00:02:57,610 --> 00:02:59,310
got two heads in that outcome.

47
00:03:02,160 --> 00:03:07,300
We could also define another
value called M, which we'll say

48
00:03:07,300 --> 00:03:16,680
is 1 if all three coins match,
if they're all the same,

49
00:03:16,680 --> 00:03:19,987
and 0 otherwise, if
they're not all the same.

50
00:03:23,400 --> 00:03:32,780
For example, M on the outcome
heads, heads, tails is 0.

51
00:03:32,780 --> 00:03:36,190
M on the outcome
tails, tails, tails

52
00:03:36,190 --> 00:03:38,120
would be 1, because
they all are the same.

53
00:03:40,930 --> 00:03:42,120
Is M a random variable?

54
00:03:45,480 --> 00:03:49,370
Yeah, it assigns a value, a
real number, to every outcome.

55
00:03:49,370 --> 00:03:52,110
And that value is
either 0 or 1, depending

56
00:03:52,110 --> 00:03:53,980
on if all the coins match.

57
00:03:53,980 --> 00:03:56,870
In fact, it's a very special
kind of random variable

58
00:03:56,870 --> 00:03:59,520
that's called an indicator
random variable, also

59
00:03:59,520 --> 00:04:03,820
known as Bernoulli or
characteristic random variable,

60
00:04:03,820 --> 00:04:05,270
because it's zero-one.

61
00:04:08,860 --> 00:04:21,209
An indicator also
known as a Bernoulli

62
00:04:21,209 --> 00:04:35,890
or characteristic
random variable,

63
00:04:35,890 --> 00:04:44,130
is a random variable whose
range or possible values

64
00:04:44,130 --> 00:04:47,910
is just 0 and 1.

65
00:04:47,910 --> 00:04:50,030
So it's a random variable
where the real numbers

66
00:04:50,030 --> 00:04:53,140
are 0, 1 that you assign
to every sample point.

67
00:04:53,140 --> 00:04:56,920
And it's called a characteristic
or indicator random variable

68
00:04:56,920 --> 00:05:00,660
because it indicates
which sample points have

69
00:05:00,660 --> 00:05:02,430
a certain property.

70
00:05:02,430 --> 00:05:05,480
In this case, all the
coins were the same.

71
00:05:05,480 --> 00:05:07,310
Or it indicates
that a sample point

72
00:05:07,310 --> 00:05:09,110
has a certain
characteristic, which

73
00:05:09,110 --> 00:05:11,742
is why it's called a
characteristic random variable.

74
00:05:11,742 --> 00:05:13,200
And we're going to
be talking a lot

75
00:05:13,200 --> 00:05:14,800
about these kinds
of random variables

76
00:05:14,800 --> 00:05:17,760
over the next couple of weeks.

77
00:05:17,760 --> 00:05:21,440
Now, in general, a random
variable is equivalent to,

78
00:05:21,440 --> 00:05:26,470
or it can define a partition
on the sample space.

79
00:05:26,470 --> 00:05:29,710
For an indicator
random variable,

80
00:05:29,710 --> 00:05:33,220
it defines a partition
with two blocks.

81
00:05:33,220 --> 00:05:38,470
For example, say we look at the
sample space with three coin

82
00:05:38,470 --> 00:05:41,080
tosses there.

83
00:05:41,080 --> 00:05:42,180
There's eight outcomes.

84
00:05:42,180 --> 00:05:50,400
There's heads, heads,
heads; heads, tails, heads;

85
00:05:50,400 --> 00:05:55,556
heads, heads, tails;
tail, heads, heads;

86
00:05:55,556 --> 00:05:56,835
and the reverse of those.

87
00:06:04,470 --> 00:06:07,180
This is the sample space
when I toss three coins.

88
00:06:07,180 --> 00:06:09,940
There's eight possible outcomes.

89
00:06:09,940 --> 00:06:12,845
The random variable M
defines this partition.

90
00:06:15,700 --> 00:06:20,790
Up top, you have M equals 1, and
down below you have M equals 0.

91
00:06:20,790 --> 00:06:23,450
Because these are the outcomes
where all the coins match.

92
00:06:23,450 --> 00:06:24,700
These are the ones that don't.

93
00:06:27,820 --> 00:06:30,775
The random variable R defines
a different partition.

94
00:06:33,740 --> 00:06:38,320
In that case, the
partition has four blocks.

95
00:06:38,320 --> 00:06:44,860
Here is R equals 0, no heads.

96
00:06:44,860 --> 00:06:48,660
This is R equals 1, to one head.

97
00:06:48,660 --> 00:06:51,340
R equals 2, and R equals 3.

98
00:06:54,460 --> 00:06:58,750
So a random variable just sort
of organizes your sample space,

99
00:06:58,750 --> 00:07:01,450
partitions it into
blocks, defined

100
00:07:01,450 --> 00:07:03,560
by all the elements
in the block,

101
00:07:03,560 --> 00:07:05,976
the outcomes in the
block have the same value

102
00:07:05,976 --> 00:07:06,975
for the random variable.

103
00:07:09,500 --> 00:07:14,000
And, in fact, every
block is really an event.

104
00:07:14,000 --> 00:07:17,040
In particular, we
can say that, if you

105
00:07:17,040 --> 00:07:21,530
look at the outcomes for
which the random variable

106
00:07:21,530 --> 00:07:24,830
on that outcome
equals some value x,

107
00:07:24,830 --> 00:07:32,160
this is simply the
event that R equals x.

108
00:07:32,160 --> 00:07:34,900
The random variable is x.

109
00:07:34,900 --> 00:07:37,490
And, of course, an event is just
a subset of the sample space.

110
00:07:37,490 --> 00:07:40,345
It's a collection of outcomes
that define the event.

111
00:07:42,890 --> 00:07:45,870
And since we know we could talk
about probabilities of events,

112
00:07:45,870 --> 00:07:47,900
we can now say things like this.

113
00:07:47,900 --> 00:07:52,610
We can say the probability
that a random variable equals

114
00:07:52,610 --> 00:07:57,950
x is simply the probability
of that event happening, which

115
00:07:57,950 --> 00:08:03,890
is the sum over all outcomes
for which the random variable

116
00:08:03,890 --> 00:08:08,850
equals x for that outcome,
summing its probability.

117
00:08:08,850 --> 00:08:12,060
So the probability that the
random variable equals a value

118
00:08:12,060 --> 00:08:14,500
is the sum of the probabilities
of the sample points

119
00:08:14,500 --> 00:08:17,500
for which R has that value.

120
00:08:17,500 --> 00:08:19,250
All right, this is all
really basic stuff,

121
00:08:19,250 --> 00:08:23,200
but important to get down.

122
00:08:23,200 --> 00:08:26,960
So, for example,
what's the probability

123
00:08:26,960 --> 00:08:31,372
that R equals 2 in our
case with the three coins?

124
00:08:31,372 --> 00:08:35,090
You flip three coins-- let's say
the coins are fair and mutually

125
00:08:35,090 --> 00:08:35,860
independent.

126
00:08:35,860 --> 00:08:38,720
We probably need to know that.

127
00:08:38,720 --> 00:08:41,675
What is the
probability R equals 2?

128
00:08:41,675 --> 00:08:42,950
AUDIENCE: 3/8?

129
00:08:42,950 --> 00:08:47,780
PROFESSOR: 3/8, because you've
got three outcomes, each

130
00:08:47,780 --> 00:08:50,050
with a probability 1/8.

131
00:08:50,050 --> 00:08:52,590
All right, so by
this definition, that

132
00:08:52,590 --> 00:08:56,640
would be the same as the
probability of heads, heads,

133
00:08:56,640 --> 00:09:01,240
tails; heads, tails,
heads; tails, heads,

134
00:09:01,240 --> 00:09:03,160
heads; that's 3/8.

135
00:09:06,760 --> 00:09:10,660
What's the probability
that M equals 1?

136
00:09:16,928 --> 00:09:19,300
What's the probability
that M equals 1?

137
00:09:19,300 --> 00:09:23,050
M is 1 if all the coins match.

138
00:09:23,050 --> 00:09:27,360
1/4 because you've got two
cases there-- heads, heads,

139
00:09:27,360 --> 00:09:31,720
heads-- whoops--
or tails, tails,

140
00:09:31,720 --> 00:09:36,982
tails; and that's 2/8, or 1/4.

141
00:09:36,982 --> 00:09:39,840
Now we can also talk
about the probability

142
00:09:39,840 --> 00:09:43,930
that a random variable
is in some range.

143
00:09:43,930 --> 00:09:47,180
For example, I could ask,
what's the probability

144
00:09:47,180 --> 00:09:49,925
that R is at least 2?

145
00:09:52,525 --> 00:09:58,340
All right, that would mean the
sum i equals 2 to infinity,

146
00:09:58,340 --> 00:10:02,230
the probability R equals i.

147
00:10:02,230 --> 00:10:03,035
And what is that?

148
00:10:05,930 --> 00:10:07,150
What's the probability?

149
00:10:07,150 --> 00:10:09,740
R is at least 2.

150
00:10:09,740 --> 00:10:10,480
1/2.

151
00:10:10,480 --> 00:10:12,469
It's the probability
that R equals 2,

152
00:10:12,469 --> 00:10:14,010
plus the probability
that R equals 3.

153
00:10:14,010 --> 00:10:16,840
There's four outcomes there,
each with 1/8, so that's 1/2.

154
00:10:21,300 --> 00:10:24,030
And finally, we can talk
about the probability

155
00:10:24,030 --> 00:10:28,770
that R is in some set of values.

156
00:10:28,770 --> 00:10:34,670
So we could say, for
example, for any subset

157
00:10:34,670 --> 00:10:38,830
of the real numbers, we
define the probability

158
00:10:38,830 --> 00:10:44,100
that R attains a value
in that set is simply

159
00:10:44,100 --> 00:10:50,660
the sum over all possible
values in the set,

160
00:10:50,660 --> 00:10:52,570
that the probability
R equals that value.

161
00:10:55,570 --> 00:11:00,170
So we could ask, what's the
probability that, say, we

162
00:11:00,170 --> 00:11:04,950
take A being the set 1, 3.

163
00:11:04,950 --> 00:11:08,120
I could ask for
the probability R

164
00:11:08,120 --> 00:11:13,380
is A when it's 1, 3, which is
the same as the probability

165
00:11:13,380 --> 00:11:15,220
R is odd.

166
00:11:15,220 --> 00:11:18,210
Because Our could
only be 0, 1, 2, or 3.

167
00:11:18,210 --> 00:11:24,494
What's the probability R
is odd in this example?

168
00:11:24,494 --> 00:11:26,410
Flip three coins, we're
asking the probability

169
00:11:26,410 --> 00:11:31,240
you get an odd number of
heads-- one or three heads.

170
00:11:31,240 --> 00:11:32,650
1/2.

171
00:11:32,650 --> 00:11:40,057
Again, you've got four sample
points-- here and there.

172
00:11:40,057 --> 00:11:40,640
So that's 1/2.

173
00:11:44,820 --> 00:11:48,125
All right, any questions
about the definition?

174
00:11:51,210 --> 00:11:54,130
All right, now, we're only
going to worry about the case

175
00:11:54,130 --> 00:11:58,100
when random variables
are discrete.

176
00:11:58,100 --> 00:12:02,940
Namely, there's a
finite number of values.

177
00:12:02,940 --> 00:12:05,190
If you have continuous random
variables, which they'll

178
00:12:05,190 --> 00:12:06,550
deal with in other
courses, instead

179
00:12:06,550 --> 00:12:08,010
of doing sums and
stuff like that,

180
00:12:08,010 --> 00:12:09,260
you're working with integrals.

181
00:12:09,260 --> 00:12:12,840
But for this course, it's
going to be all countable sets,

182
00:12:12,840 --> 00:12:16,770
and usually often finite sets.

183
00:12:16,770 --> 00:12:21,300
Now, conditional probability
carries over very nicely

184
00:12:21,300 --> 00:12:22,690
to random variables as well.

185
00:12:22,690 --> 00:12:27,160
For example, we could
write down the probability

186
00:12:27,160 --> 00:12:33,974
that R equals 2 given that M
equals 1, where R and M are

187
00:12:33,974 --> 00:12:36,140
the same random variables
we've been looking at when

188
00:12:36,140 --> 00:12:38,340
you flip the three coins.

189
00:12:38,340 --> 00:12:40,830
What is that probability?

190
00:12:40,830 --> 00:12:43,960
0, the probability of getting
exactly two heads when they're

191
00:12:43,960 --> 00:12:46,210
all the same, can't happen.

192
00:12:46,210 --> 00:12:46,990
It's 0.

193
00:12:46,990 --> 00:12:50,590
You can either have
three heads or no heads.

194
00:12:50,590 --> 00:12:51,380
So that's 0.

195
00:12:54,920 --> 00:12:57,852
The notion of independence
also carries over

196
00:12:57,852 --> 00:12:59,310
to random variables,
but you've got

197
00:12:59,310 --> 00:13:00,530
to be a little careful here.

198
00:13:06,670 --> 00:13:12,870
Two random variables,
R1 and R2, are

199
00:13:12,870 --> 00:13:26,070
said to be independent if-- and
this is a little complicated--

200
00:13:26,070 --> 00:13:31,392
for all possible
values, x1 and x2

201
00:13:31,392 --> 00:13:40,290
in the real numbers, the
probability that R1 is x1,

202
00:13:40,290 --> 00:13:46,580
given that R2 is x2, is the
same as the probability of R1

203
00:13:46,580 --> 00:13:48,910
equals x1 not knowing
anything about R2.

204
00:13:54,060 --> 00:13:57,610
Or there's a special case
when this can't happen.

205
00:13:57,610 --> 00:14:02,190
Namely, the probability
that R2 equals x2 is 0.

206
00:14:05,020 --> 00:14:08,830
So, for two random
variables to be independent,

207
00:14:08,830 --> 00:14:12,720
it needs to be the case that,
no matter what happens out here

208
00:14:12,720 --> 00:14:15,910
with R2, and whatever
you know about it,

209
00:14:15,910 --> 00:14:19,740
it can't influence
the probability for R1

210
00:14:19,740 --> 00:14:22,030
to equal anything.

211
00:14:22,030 --> 00:14:27,790
So no value of R2
influences any value of R1.

212
00:14:27,790 --> 00:14:31,180
So it's the strongest possible
thing with independence

213
00:14:31,180 --> 00:14:33,946
for it to apply to
random variables.

214
00:14:33,946 --> 00:14:36,570
And there's an
equivalent definition,

215
00:14:36,570 --> 00:14:39,260
and we're going to use these
interchangeably, just as there

216
00:14:39,260 --> 00:14:43,010
were two definitions for
independence of events,

217
00:14:43,010 --> 00:14:45,840
there's the product for them.

218
00:14:45,840 --> 00:14:48,890
So the equivalent
definitions says

219
00:14:48,890 --> 00:14:55,500
for all x1 and x2 in the
real numbers, the probability

220
00:14:55,500 --> 00:15:02,550
that R1 equals x1 and
R2 equals x2 is simply

221
00:15:02,550 --> 00:15:05,680
the product of
those probabilities.

222
00:15:05,680 --> 00:15:12,410
Probability R1 equals x1 times
the probability R2 equals x2.

223
00:15:15,029 --> 00:15:16,820
And you don't have to
worry about that case

224
00:15:16,820 --> 00:15:19,980
where R2 being x2 is 0.

225
00:15:19,980 --> 00:15:24,490
So we can use both of those
as definitions of independence

226
00:15:24,490 --> 00:15:27,550
of random variables.

227
00:15:27,550 --> 00:15:29,430
All right, so let's
see an example.

228
00:15:29,430 --> 00:15:33,500
We've got R and M. We've talked
about two random variables.

229
00:15:33,500 --> 00:15:36,320
Are they independent
random variables?

230
00:15:40,610 --> 00:15:42,790
M is the indicated
random variable

231
00:15:42,790 --> 00:15:44,630
for all the coins matching.

232
00:15:44,630 --> 00:15:47,180
R is the random variable
that tells us how many heads

233
00:15:47,180 --> 00:15:49,210
there were in three coins.

234
00:15:49,210 --> 00:15:50,790
Are they independent?

235
00:15:50,790 --> 00:15:51,510
AUDIENCE: No.

236
00:15:51,510 --> 00:15:54,024
PROFESSOR: No, why not?

237
00:15:54,024 --> 00:15:57,293
AUDIENCE: Because information
with what one is narrows down

238
00:15:57,293 --> 00:15:57,934
[INAUDIBLE].

239
00:15:57,934 --> 00:15:59,100
PROFESSOR: Yes, that's true.

240
00:15:59,100 --> 00:16:01,560
Information about
what one value is

241
00:16:01,560 --> 00:16:04,210
influences the probability
that the other guy

242
00:16:04,210 --> 00:16:05,240
has a certain value.

243
00:16:05,240 --> 00:16:09,030
In particular, we can
find a case of x1 and x2

244
00:16:09,030 --> 00:16:11,020
where this fails.

245
00:16:11,020 --> 00:16:12,910
One case would be
the probability

246
00:16:12,910 --> 00:16:21,036
of what we've already done,
that R equals 2 and M equals 1.

247
00:16:21,036 --> 00:16:23,930
What was this probability?

248
00:16:23,930 --> 00:16:24,650
0.

249
00:16:24,650 --> 00:16:26,780
If everything matches,
you can't have two heads.

250
00:16:26,780 --> 00:16:28,110
That's 0.

251
00:16:28,110 --> 00:16:31,220
That does not equal
the probability

252
00:16:31,220 --> 00:16:36,580
of two heads times the
probability they all match.

253
00:16:36,580 --> 00:16:40,550
Because this probability,
two heads is 3/8,

254
00:16:40,550 --> 00:16:43,045
probability they
all match is 1/4.

255
00:16:43,045 --> 00:16:47,510
And 3/8 times a 1/4 is not 0.

256
00:16:47,510 --> 00:16:49,730
So R and M are not independent.

257
00:16:57,470 --> 00:17:00,560
OK, now in general to
show two random variables

258
00:17:00,560 --> 00:17:03,070
are not independent,
all you gotta

259
00:17:03,070 --> 00:17:08,180
do is find one pair of [? out ?]
values for which there's

260
00:17:08,180 --> 00:17:09,700
dependence.

261
00:17:09,700 --> 00:17:12,060
To show their
independent, you've

262
00:17:12,060 --> 00:17:15,390
got to deal with all
possible pairs of values.

263
00:17:15,390 --> 00:17:17,480
So it's harder show
independence in general.

264
00:17:20,720 --> 00:17:24,588
All right, let's
do another example,

265
00:17:24,588 --> 00:17:26,569
get a little practice with this.

266
00:17:38,180 --> 00:17:51,730
Say we have two fair independent
six-sided die, normal dice.

267
00:17:51,730 --> 00:17:54,590
And the outcome of
the first one is D1,

268
00:17:54,590 --> 00:17:58,710
and the outcome of
the second roll is D2.

269
00:17:58,710 --> 00:18:04,170
And by independent, I mean here
that knowing any information

270
00:18:04,170 --> 00:18:06,040
about what the second
die is does not

271
00:18:06,040 --> 00:18:09,020
give you any change
in your probability

272
00:18:09,020 --> 00:18:12,690
that the first
die has any value.

273
00:18:12,690 --> 00:18:15,500
And now we define
another value, S,

274
00:18:15,500 --> 00:18:19,825
to be D1 plus D2,
the sum of the dice.

275
00:18:22,490 --> 00:18:23,930
Is S a random variable?

276
00:18:27,920 --> 00:18:32,190
Yeah, for any outcome-- an
outcome being a pair of dice

277
00:18:32,190 --> 00:18:36,026
values-- it maps that
outcome to a real number,

278
00:18:36,026 --> 00:18:41,700
a number between 2 and 12, OK?

279
00:18:41,700 --> 00:18:43,560
Let's do another one.

280
00:18:43,560 --> 00:18:53,870
Let's let T be 1 if S is 7,
namely the sum of the dice

281
00:18:53,870 --> 00:18:55,200
is 7, and 0 otherwise.

282
00:18:59,860 --> 00:19:02,340
T is also a random variable.

283
00:19:02,340 --> 00:19:05,030
In fact, what kind is it?

284
00:19:05,030 --> 00:19:08,550
Yeah, indicator,
characteristic, whatever.

285
00:19:08,550 --> 00:19:13,830
It's telling you if the sum
of the dice is 7 or not.

286
00:19:13,830 --> 00:19:18,300
All right, now, there's
four random variables here.

287
00:19:18,300 --> 00:19:21,440
Each die, which we
already have told you,

288
00:19:21,440 --> 00:19:23,370
we're assuming are independent.

289
00:19:23,370 --> 00:19:26,830
The sum and the indicator
if the sum is 7.

290
00:19:30,130 --> 00:19:34,990
All right, what about
these two values?

291
00:19:34,990 --> 00:19:38,170
Are D1, the first die,
and the sum independent?

292
00:19:41,603 --> 00:19:43,998
AUDIENCE: No?

293
00:19:43,998 --> 00:19:45,440
No?

294
00:19:45,440 --> 00:19:47,850
PROFESSOR: No,
intuitively they're not.

295
00:19:47,850 --> 00:19:52,020
Because if I know something
about the first die,

296
00:19:52,020 --> 00:19:55,710
I probably know
something about the sum.

297
00:19:55,710 --> 00:19:59,110
Now to really nail
that down, you've

298
00:19:59,110 --> 00:20:02,670
got to give me a
value for this, which

299
00:20:02,670 --> 00:20:07,419
influences the probability
that obtains some value, right?

300
00:20:07,419 --> 00:20:09,710
That's how you would convince
me that, in fact, they're

301
00:20:09,710 --> 00:20:10,209
dependent.

302
00:20:13,600 --> 00:20:15,870
Can anybody give
me a value for this

303
00:20:15,870 --> 00:20:18,770
that changes the probability
this equals something?

304
00:20:23,320 --> 00:20:24,250
AUDIENCE: Guess?

305
00:20:24,250 --> 00:20:27,330
If it's like 1, then
[INAUDIBLE] has to be 6.

306
00:20:27,330 --> 00:20:32,170
PROFESSOR: If this is
1, what value of this

307
00:20:32,170 --> 00:20:35,028
could be influenced?

308
00:20:35,028 --> 00:20:39,930
AUDIENCE: You know that in order
for-- the only set of values

309
00:20:39,930 --> 00:20:44,348
S can take now, it can only
be from 1 to-- from 2 to 7.

310
00:20:44,348 --> 00:20:45,306
PROFESSOR: From 2 to 7.

311
00:20:45,306 --> 00:20:49,230
In particular now,
this can't be 12.

312
00:20:49,230 --> 00:20:51,570
So what I could do is say
the probability that S

313
00:20:51,570 --> 00:20:58,780
equals 12 and the first die
was 1, what's that probability?

314
00:20:58,780 --> 00:20:59,280
AUDIENCE: 0.

315
00:20:59,280 --> 00:21:01,300
PROFESSOR: 0.

316
00:21:01,300 --> 00:21:03,980
And that does not
equal the probability

317
00:21:03,980 --> 00:21:08,380
S is 12 times the probability
the first die was 1.

318
00:21:11,240 --> 00:21:13,690
Or we could plug it into
the other definition.

319
00:21:13,690 --> 00:21:18,290
The probability that S equals
12 given the first die is 1

320
00:21:18,290 --> 00:21:21,440
is not equal to the
probability S equals 12.

321
00:21:21,440 --> 00:21:24,510
And, in fact, probability
the first die is 1 is not 0.

322
00:21:24,510 --> 00:21:26,750
So either definition,
of course, works

323
00:21:26,750 --> 00:21:29,630
to show that they are dependent.

324
00:21:29,630 --> 00:21:31,350
So S and D1 are dependent.

325
00:21:35,800 --> 00:21:38,430
All you need is
one possible pair

326
00:21:38,430 --> 00:21:40,360
of values to show dependence.

327
00:21:40,360 --> 00:21:41,660
That's usually easy to do.

328
00:21:46,350 --> 00:21:52,700
All right, what about
D1, the first die, and T,

329
00:21:52,700 --> 00:21:55,695
the indicator for getting a 7?

330
00:21:59,150 --> 00:22:03,310
Are D1 and T independent
random variables?

331
00:22:05,815 --> 00:22:08,860
All right, somebody's
shaking their head no.

332
00:22:08,860 --> 00:22:12,430
They seem to be dependent
because knowing the first die

333
00:22:12,430 --> 00:22:15,660
seems like it should tell you
something about the probability

334
00:22:15,660 --> 00:22:19,930
of getting a 7 for the sum.

335
00:22:19,930 --> 00:22:22,140
That's a good first intuition.

336
00:22:22,140 --> 00:22:23,770
In fact, you always
want to assume

337
00:22:23,770 --> 00:22:27,070
things are dependent unless you
convince yourself otherwise.

338
00:22:27,070 --> 00:22:29,770
It's always a good rule.

339
00:22:29,770 --> 00:22:34,200
Now, in fact, in this
case, it is independent.

340
00:22:34,200 --> 00:22:37,910
The probability of getting
a 7 is not, it turns out,

341
00:22:37,910 --> 00:22:41,540
influenced by the first die.

342
00:22:41,540 --> 00:22:44,380
Anything else would
be, but 7 is not.

343
00:22:44,380 --> 00:22:45,827
Let's see why that is.

344
00:22:45,827 --> 00:22:47,660
And to do that, we'd
actually have to check,

345
00:22:47,660 --> 00:22:51,700
I think, 12 cases, all
possible values for D1 and T,

346
00:22:51,700 --> 00:22:54,710
but you'll get the
idea pretty quick.

347
00:22:54,710 --> 00:22:59,620
The probability that T
equals 1, namely you got a 7,

348
00:22:59,620 --> 00:23:00,970
given that D1 equals 1.

349
00:23:00,970 --> 00:23:03,180
What's that?

350
00:23:03,180 --> 00:23:06,965
The probability of getting 7
for the sum given your first die

351
00:23:06,965 --> 00:23:08,760
is a 1, what's that probability?

352
00:23:08,760 --> 00:23:09,540
AUDIENCE: 1/6?

353
00:23:09,540 --> 00:23:13,730
PROFESSOR: 1/6, because the
second one better be a 6.

354
00:23:13,730 --> 00:23:19,500
All right, and what's the
probability T equals 1?

355
00:23:19,500 --> 00:23:21,425
What's the probability
of rolling a 7?

356
00:23:24,810 --> 00:23:26,096
Yeah?

357
00:23:26,096 --> 00:23:27,240
AUDIENCE: 6/36.

358
00:23:27,240 --> 00:23:28,670
PROFESSOR: 6/36.

359
00:23:28,670 --> 00:23:30,160
There's six ways to do it.

360
00:23:30,160 --> 00:23:35,880
1 and 6, 2 and 5, 3 and 4, 4
and 3, 5 and 2, and 6 and 1,

361
00:23:35,880 --> 00:23:39,370
out of 36 possibilities
equally likely.

362
00:23:39,370 --> 00:23:42,670
So that worked here.

363
00:23:42,670 --> 00:23:46,240
Now probability of getting
a 7 given the second

364
00:23:46,240 --> 00:23:49,605
die-- sorry-- the
first die is a 2.

365
00:23:49,605 --> 00:23:50,105
What's that?

366
00:23:53,740 --> 00:23:58,310
Probability of getting a 7 given
that your first die is a 2.

367
00:23:58,310 --> 00:24:02,720
1/6, because you've got to
have the second die be a 5,

368
00:24:02,720 --> 00:24:05,670
and that happens 1/6.

369
00:24:05,670 --> 00:24:08,740
And that equals the right thing.

370
00:24:08,740 --> 00:24:11,800
In fact, I can
keep on going here.

371
00:24:11,800 --> 00:24:18,560
Probability T equals 1,
given D1 is anything, 6.

372
00:24:18,560 --> 00:24:20,450
Well, for every
value of D1, there's

373
00:24:20,450 --> 00:24:25,690
exactly one value of
D2 that adds to a 7.

374
00:24:25,690 --> 00:24:31,060
So it's still 1/6, which is
the probability of getting

375
00:24:31,060 --> 00:24:33,770
the 7 in the first place.

376
00:24:33,770 --> 00:24:38,700
Now we've also got to check
the probability T equals 0,

377
00:24:38,700 --> 00:24:40,910
given all these cases for D1.

378
00:24:40,910 --> 00:24:43,340
D1 equals 1.

379
00:24:43,340 --> 00:24:43,850
What's this?

380
00:24:43,850 --> 00:24:46,970
The probability of not getting
a 7 given the first die is a 1.

381
00:24:46,970 --> 00:24:49,875
What's that?

382
00:24:49,875 --> 00:24:50,780
AUDIENCE: 5/6.

383
00:24:50,780 --> 00:24:53,820
PROFESSOR: 5/6, it's the
opposite of this case.

384
00:24:53,820 --> 00:24:54,460
They pair up.

385
00:24:54,460 --> 00:24:59,300
That's got to be 5/6,
and that is, of course,

386
00:24:59,300 --> 00:25:02,960
the probability of T equals 0.

387
00:25:02,960 --> 00:25:07,810
And the same is true for
all the other five cases.

388
00:25:07,810 --> 00:25:14,200
The probability T equals
0 given any value of D

389
00:25:14,200 --> 00:25:19,910
is 5/6, which equals, the
probability T equals 0.

390
00:25:19,910 --> 00:25:22,860
So we check all
possible values of T1

391
00:25:22,860 --> 00:25:25,710
and all possible ways of
D1, and lo and behold,

392
00:25:25,710 --> 00:25:28,050
it works every time.

393
00:25:28,050 --> 00:25:29,840
So knowing the
first die does not

394
00:25:29,840 --> 00:25:33,630
change the probability
of getting a 7.

395
00:25:33,630 --> 00:25:35,595
Would it change the
probability of getting a 6?

396
00:25:38,150 --> 00:25:41,040
Yeah, because if the
first die were a 6,

397
00:25:41,040 --> 00:25:42,960
you know you're going
to get more than a 6.

398
00:25:42,960 --> 00:25:46,000
So it just holds
true because I picked

399
00:25:46,000 --> 00:25:49,220
it to be an indicator for 7.

400
00:25:49,220 --> 00:25:52,780
Any questions about that?

401
00:25:52,780 --> 00:25:56,900
So if you're ever asked to
show things are independent,

402
00:25:56,900 --> 00:25:59,424
you've got to go
through all the cases.

403
00:25:59,424 --> 00:26:01,090
If you're asked to
show their dependent,

404
00:26:01,090 --> 00:26:06,830
just find one case that
blows it up, and you're done.

405
00:26:06,830 --> 00:26:10,090
OK, you can also talk
about independence

406
00:26:10,090 --> 00:26:12,466
for many random variables.

407
00:26:12,466 --> 00:26:14,340
And you have a notion
of mutual independence.

408
00:26:26,900 --> 00:26:30,080
And this is a little hairy.

409
00:26:30,080 --> 00:26:32,090
It's a natural thing.

410
00:26:32,090 --> 00:26:54,400
R1, R2, up to Rn are mutually
independent if for all

411
00:26:54,400 --> 00:27:00,430
the values of all the random
variables, so x1, x2, xn.

412
00:27:03,941 --> 00:27:06,190
And we're going to give the
product form, because it's

413
00:27:06,190 --> 00:27:08,650
the simpler way to do it here.

414
00:27:08,650 --> 00:27:19,170
The probability that R1 is
x1, and R2 is x2, and Rn is xn

415
00:27:19,170 --> 00:27:21,450
equals the product of the
individual probabilities.

416
00:27:30,880 --> 00:27:35,420
It's the natural generalization
for two variables.

417
00:27:35,420 --> 00:27:37,950
And there's also
the equivalent form

418
00:27:37,950 --> 00:27:40,665
in conditional probabilities,
but that's a little harrier

419
00:27:40,665 --> 00:27:43,710
to write down, and we
generally don't work with that.

420
00:27:49,880 --> 00:27:54,340
Any questions about
mutual independence?

421
00:27:54,340 --> 00:27:55,227
Yeah.

422
00:27:55,227 --> 00:27:56,143
AUDIENCE: [INAUDIBLE].

423
00:28:04,173 --> 00:28:05,664
PROFESSOR: So any subset what?

424
00:28:05,664 --> 00:28:06,580
AUDIENCE: [INAUDIBLE].

425
00:28:09,860 --> 00:28:11,720
PROFESSOR: Yes, in
the conditional form

426
00:28:11,720 --> 00:28:15,770
you check any subset,
here you don't have

427
00:28:15,770 --> 00:28:16,940
to worry about the subsets.

428
00:28:16,940 --> 00:28:20,550
You take any possible
value for all of them,

429
00:28:20,550 --> 00:28:24,290
so there's no subset
notation in this version,

430
00:28:24,290 --> 00:28:25,706
so you can do without it.

431
00:28:25,706 --> 00:28:27,080
That would be
equivalent, there's

432
00:28:27,080 --> 00:28:29,371
another form where you look
at all the possible subsets

433
00:28:29,371 --> 00:28:31,030
as well, and
conditioning on that.

434
00:28:31,030 --> 00:28:32,932
But this is good enough,
and this is probably

435
00:28:32,932 --> 00:28:34,890
the simplest way to do
the mutual independence,

436
00:28:34,890 --> 00:28:36,150
is like this.

437
00:28:36,150 --> 00:28:38,850
So every variable
is included here.

438
00:28:38,850 --> 00:28:41,950
And that will imply the
same thing for any subset.

439
00:28:41,950 --> 00:28:45,380
You don't have to check
every subset here.

440
00:28:45,380 --> 00:28:47,620
And then you could do
that by just summing

441
00:28:47,620 --> 00:28:51,560
over Rn being everything,
and that cancels away

442
00:28:51,560 --> 00:28:53,016
to get rid of items.

443
00:28:53,016 --> 00:28:54,640
But you don't have
to worry about that.

444
00:28:58,020 --> 00:29:00,540
OK, so we're going to
change gears a little bit

445
00:29:00,540 --> 00:29:06,990
now, and talk about the
probability distribution

446
00:29:06,990 --> 00:29:10,600
function, which is just
a way a sort of writing

447
00:29:10,600 --> 00:29:14,860
down or characterizing the
probabilities associated

448
00:29:14,860 --> 00:29:16,625
with each value being attained.

449
00:29:21,920 --> 00:29:33,690
So given a random variable,
R, the probability

450
00:29:33,690 --> 00:29:53,360
also known as the point,
distribution function,

451
00:29:53,360 --> 00:30:01,580
also denoted pdf, probability
distribution function, for R

452
00:30:01,580 --> 00:30:03,700
is, well, it's very simple.

453
00:30:03,700 --> 00:30:07,940
It's just the function
f of x, which equals

454
00:30:07,940 --> 00:30:11,320
the probability that R is x.

455
00:30:11,320 --> 00:30:12,940
Very simple.

456
00:30:12,940 --> 00:30:17,170
But now we're characterizing
it as a function.

457
00:30:17,170 --> 00:30:20,320
And there's also a notion of
the cumulative distribution

458
00:30:20,320 --> 00:30:24,830
function, which is going to
be the probability that R

459
00:30:24,830 --> 00:30:27,130
is less than or equal to x.

460
00:30:27,130 --> 00:30:28,231
Let's write that down.

461
00:30:52,800 --> 00:31:12,500
The cumulative
distribution function F

462
00:31:12,500 --> 00:31:19,820
for random variable
R is simply f

463
00:31:19,820 --> 00:31:25,840
of x is the probability that
R is at most x, which is just

464
00:31:25,840 --> 00:31:30,730
the sum over all y
less than or equal to x

465
00:31:30,730 --> 00:31:32,510
the probability R equals y.

466
00:31:35,480 --> 00:31:40,930
So the distribution functions
characterize the probability

467
00:31:40,930 --> 00:31:43,560
a random variable
takes on any value,

468
00:31:43,560 --> 00:31:47,400
and there's certain common ones
that just come up all the time.

469
00:31:47,400 --> 00:31:49,190
And so people have
done a lot of analysis

470
00:31:49,190 --> 00:31:52,990
about them because they
occur so frequently.

471
00:31:52,990 --> 00:31:56,110
And we're going to talk
about three of them today.

472
00:32:02,830 --> 00:32:06,860
The first is really
simple, and that

473
00:32:06,860 --> 00:32:10,000
is the Bernoulli
random variable,

474
00:32:10,000 --> 00:32:11,820
or indicator random variable.

475
00:32:14,430 --> 00:32:29,520
All right, so for an indicator
random variable, f 0 is p,

476
00:32:29,520 --> 00:32:36,290
and f 1 is 1 minus p, for sum p.

477
00:32:36,290 --> 00:32:38,817
The probability could be half,
if you're flipping a coin.

478
00:32:38,817 --> 00:32:41,400
The probability of heads could
be a half, probability of tails

479
00:32:41,400 --> 00:32:42,066
would be a half.

480
00:32:42,066 --> 00:32:45,360
It's just two values, 0 and 1.

481
00:32:45,360 --> 00:32:48,770
And then the cumulative
function is very simple.

482
00:32:48,770 --> 00:32:53,130
F 0 is p, big F of 1 is 1.

483
00:32:55,710 --> 00:32:58,225
So that is about as simple
of functions as you can get.

484
00:33:03,670 --> 00:33:07,390
The next simplest,
and also very common,

485
00:33:07,390 --> 00:33:10,370
is called a uniform
random variable.

486
00:33:14,970 --> 00:33:16,380
Let's define that.

487
00:33:26,300 --> 00:33:32,610
For a uniform random
variable-- and we

488
00:33:32,610 --> 00:33:35,000
have to define what
it's defined on--

489
00:33:35,000 --> 00:33:42,900
on say the integers from 1 to n,
every value is equally likely.

490
00:33:42,900 --> 00:33:46,600
All the integers from 1 to n
are equally likely to occur.

491
00:33:46,600 --> 00:33:51,470
And so in this case, n is a
parameter of the function.

492
00:33:51,470 --> 00:33:54,750
fn of K is simply 1/n.

493
00:33:54,750 --> 00:33:57,860
Each integer from 1 to
n is equally likely.

494
00:33:57,860 --> 00:33:59,260
So it's 1 and n.

495
00:34:03,990 --> 00:34:06,360
All right, what is the
cumulative distribution

496
00:34:06,360 --> 00:34:08,920
function for this?

497
00:34:08,920 --> 00:34:13,989
What is big F of n
of K for the uniform?

498
00:34:13,989 --> 00:34:18,350
The probability that
the random variable

499
00:34:18,350 --> 00:34:22,518
takes a value that's at most K?

500
00:34:22,518 --> 00:34:24,449
AUDIENCE: [INAUDIBLE]?

501
00:34:24,449 --> 00:34:27,199
PROFESSOR: Close, K/n.

502
00:34:27,199 --> 00:34:30,139
And you could
think of K being 1.

503
00:34:30,139 --> 00:34:33,469
You have a least the
1 over n probability.

504
00:34:33,469 --> 00:34:35,290
So you have K chances.

505
00:34:35,290 --> 00:34:40,159
It could be 1, 2, 3, 4 up
to K, each has a 1/n chance.

506
00:34:40,159 --> 00:34:42,380
Because we've got,
the definition

507
00:34:42,380 --> 00:34:47,469
is less than or
equal, not less than.

508
00:34:51,020 --> 00:34:53,250
Uniform distributions
come up all the time,

509
00:34:53,250 --> 00:34:59,460
rolling a fair die is uniform
on 1 through 6 for values.

510
00:34:59,460 --> 00:35:02,560
If I pick a random student
in the class, the chance

511
00:35:02,560 --> 00:35:05,940
I pick you is 1 in
the size of the class.

512
00:35:05,940 --> 00:35:07,520
We have uniform sample spaces.

513
00:35:07,520 --> 00:35:12,300
Each outcome occurs
equally likely.

514
00:35:12,300 --> 00:35:16,340
All right, any
questions about uniform?

515
00:35:16,340 --> 00:35:18,552
[INAUDIBLE]

516
00:35:18,552 --> 00:35:20,260
OK, so while we're
talking about uniform,

517
00:35:20,260 --> 00:35:24,970
I want to play a
game, whose optimal

518
00:35:24,970 --> 00:35:26,980
strategy is going to
turn out to be related

519
00:35:26,980 --> 00:35:30,120
to uniform distributions.

520
00:35:30,120 --> 00:35:33,750
Now the game works as follows.

521
00:35:33,750 --> 00:35:37,890
I'm going to have
two envelopes here.

522
00:35:37,890 --> 00:35:42,400
And inside each envelope,
I've written a number.

523
00:35:42,400 --> 00:35:45,060
And the number in
this case is a number

524
00:35:45,060 --> 00:35:47,155
between 0 and 100 inclusive.

525
00:35:50,710 --> 00:35:54,090
It's not a random number,
because I wrote it,

526
00:35:54,090 --> 00:35:56,600
and maybe I'm not
such a nice guy.

527
00:35:56,600 --> 00:35:59,230
I might have picked nasty
numbers to write here.

528
00:35:59,230 --> 00:36:00,910
They are different, though.

529
00:36:00,910 --> 00:36:04,060
The numbers are different
in the envelopes.

530
00:36:04,060 --> 00:36:07,260
Now I'm going to pick a
volunteer from the class,

531
00:36:07,260 --> 00:36:10,630
and their job is to pick
the envelope with the bigger

532
00:36:10,630 --> 00:36:12,200
number.

533
00:36:12,200 --> 00:36:14,910
And if they get the one
with the bigger number,

534
00:36:14,910 --> 00:36:19,520
then they get one of
these guys for ice cream.

535
00:36:19,520 --> 00:36:22,910
And if they don't get the
one with the bigger number,

536
00:36:22,910 --> 00:36:27,340
they get the nerd
pride pocket protector.

537
00:36:27,340 --> 00:36:31,030
Now, since you don't know
which number I put in which,

538
00:36:31,030 --> 00:36:35,520
50-50 if you get the prize,
the $10 gift certificate.

539
00:36:35,520 --> 00:36:37,980
So to give you a
little bit of an edge,

540
00:36:37,980 --> 00:36:41,060
I'm going to let you
open one of the envelopes

541
00:36:41,060 --> 00:36:43,320
and see what number's there.

542
00:36:43,320 --> 00:36:46,780
Then you have to decide, do
you want the number you got,

543
00:36:46,780 --> 00:36:50,210
or do you want the
other envelope?

544
00:36:50,210 --> 00:36:52,330
OK?

545
00:36:52,330 --> 00:36:54,850
Is the game clear,
what we're going to do?

546
00:36:54,850 --> 00:36:56,740
OK, so need a couple
of volunteers.

547
00:36:56,740 --> 00:36:58,950
We're going to do it twice.

548
00:36:58,950 --> 00:37:01,590
All right, somebody way
in the back up there.

549
00:37:01,590 --> 00:37:02,520
Any other volunteers?

550
00:37:02,520 --> 00:37:03,520
All right, come on down.

551
00:37:03,520 --> 00:37:05,017
Oh, you've already
played before.

552
00:37:05,017 --> 00:37:07,190
(LAUGHTER) You already got.

553
00:37:07,190 --> 00:37:12,510
Who hasn't played before, wants
to-- all right, come on up.

554
00:37:12,510 --> 00:37:15,641
Now you want to be thinking
about, boy, is there a strategy

555
00:37:15,641 --> 00:37:16,140
here?

556
00:37:16,140 --> 00:37:17,660
Or is this just
dumb luck, 50-50?

557
00:37:20,960 --> 00:37:21,965
OK, what's your name?

558
00:37:21,965 --> 00:37:22,590
AUDIENCE: Sean.

559
00:37:22,590 --> 00:37:23,410
PROFESSOR: Sean.

560
00:37:23,410 --> 00:37:25,700
All right, I got
two envelopes, Sean.

561
00:37:25,700 --> 00:37:28,170
They are numbers
between 0 and 100.

562
00:37:28,170 --> 00:37:33,707
And you can pick one, and
we'll reveal the number,

563
00:37:33,707 --> 00:37:34,540
and then you decide.

564
00:37:34,540 --> 00:37:37,081
Do you want the number you got,
or do you want the other one?

565
00:37:37,081 --> 00:37:39,510
The goal is to get
the bigger number.

566
00:37:39,510 --> 00:37:40,650
So take one and open it.

567
00:37:48,105 --> 00:37:49,099
What did you get?

568
00:37:49,099 --> 00:37:50,587
AUDIENCE: 6.

569
00:37:50,587 --> 00:37:51,087
He got

570
00:37:51,087 --> 00:37:52,590
PROFESSOR: A 6.

571
00:37:52,590 --> 00:37:56,070
The numbers go from 0 to 100.

572
00:37:56,070 --> 00:37:57,880
What do you think
Sean should do?

573
00:37:57,880 --> 00:37:59,202
AUDIENCE: Switch.

574
00:37:59,202 --> 00:37:59,701
[INAUDIBLE]

575
00:38:02,250 --> 00:38:04,289
PROFESSOR: What do you
think you should do?

576
00:38:04,289 --> 00:38:05,205
AUDIENCE: [INAUDIBLE].

577
00:38:08,670 --> 00:38:10,650
AUDIENCE: They've got
to check the other one.

578
00:38:10,650 --> 00:38:13,085
PROFESSOR: No, no, you
can't look at the other one.

579
00:38:13,085 --> 00:38:14,710
And unfortunately,
you're going to play

580
00:38:14,710 --> 00:38:16,850
with different envelopes.

581
00:38:16,850 --> 00:38:18,240
What should you do?

582
00:38:18,240 --> 00:38:22,410
6, there's a lot of numbers
bigger than 6, Sean.

583
00:38:22,410 --> 00:38:25,920
But they might not
be in that envelope.

584
00:38:25,920 --> 00:38:27,730
Might be a 0 in that envelope.

585
00:38:27,730 --> 00:38:28,900
AUDIENCE: That would suck.

586
00:38:28,900 --> 00:38:29,180
PROFESSOR: Yeah.

587
00:38:29,180 --> 00:38:30,110
AUDIENCE: I'll stay.

588
00:38:30,110 --> 00:38:31,220
PROFESSOR: You're
going to stay with 6?

589
00:38:31,220 --> 00:38:31,930
AUDIENCE: Yup.

590
00:38:31,930 --> 00:38:32,680
PROFESSOR: All
right, he picked 6.

591
00:38:32,680 --> 00:38:34,263
What do you think
he should have done?

592
00:38:34,263 --> 00:38:36,350
How many people think
he should have switched?

593
00:38:36,350 --> 00:38:37,210
Ooh, Sean.

594
00:38:37,210 --> 00:38:40,010
How many people
like Sean's choice?

595
00:38:40,010 --> 00:38:40,700
Not so good.

596
00:38:40,700 --> 00:38:42,033
All right, let's see if you won.

597
00:38:45,470 --> 00:38:46,855
Here we go.

598
00:38:46,855 --> 00:38:48,550
5.

599
00:38:48,550 --> 00:38:50,560
Sean wins the ice cream.

600
00:38:50,560 --> 00:38:52,150
Good work.

601
00:38:52,150 --> 00:38:56,147
Now Sean, what was
your thinking here?

602
00:38:56,147 --> 00:38:58,582
AUDIENCE: I was
thinking [INAUDIBLE]

603
00:38:58,582 --> 00:39:03,452
in the other envelope,
knowing that 6 doesn't tell me

604
00:39:03,452 --> 00:39:05,887
anything about what's
in the other one,

605
00:39:05,887 --> 00:39:07,835
so my chances of
seeing [INAUDIBLE].

606
00:39:10,760 --> 00:39:12,900
PROFESSOR: Totally 50-50.

607
00:39:12,900 --> 00:39:15,130
How many people
buy that argument?

608
00:39:15,130 --> 00:39:15,756
It's 50-50.

609
00:39:15,756 --> 00:39:17,880
He really has no idea what's
in the other envelope.

610
00:39:17,880 --> 00:39:21,160
How many people think
there's a better way?

611
00:39:21,160 --> 00:39:21,785
Not too many.

612
00:39:21,785 --> 00:39:23,410
All right, we're
going to try it again.

613
00:39:23,410 --> 00:39:24,170
Well done.

614
00:39:24,170 --> 00:39:26,304
I've got different
envelopes here.

615
00:39:26,304 --> 00:39:27,470
And tell me your name again?

616
00:39:27,470 --> 00:39:28,095
AUDIENCE: Drew.

617
00:39:28,095 --> 00:39:29,510
PROFESSOR: Drew, OK Drew.

618
00:39:29,510 --> 00:39:31,230
Two envelopes.

619
00:39:31,230 --> 00:39:33,136
Which one would
you like to open?

620
00:39:33,136 --> 00:39:34,510
AUDIENCE: They're
both labeled B.

621
00:39:34,510 --> 00:39:37,430
PROFESSOR: They're both labeled
B. That won't help you there.

622
00:39:37,430 --> 00:39:38,430
All right, he's got one.

623
00:39:41,600 --> 00:39:42,570
Let's see what you got.

624
00:39:49,510 --> 00:39:51,620
92.

625
00:39:51,620 --> 00:39:54,520
Oh my goodness, and
we've seen what 5 and 6.

626
00:39:54,520 --> 00:39:55,830
What are you going to do?

627
00:39:55,830 --> 00:39:57,300
92.

628
00:39:57,300 --> 00:39:58,540
What should he do, guys?

629
00:39:58,540 --> 00:39:59,174
AUDIENCE: Stay.

630
00:39:59,174 --> 00:39:59,840
PROFESSOR: Stay.

631
00:39:59,840 --> 00:40:02,780
Oh he's got a big number there.

632
00:40:02,780 --> 00:40:04,160
You're going to switch?

633
00:40:04,160 --> 00:40:06,670
You're giving up a 92?

634
00:40:06,670 --> 00:40:09,480
Ooh, you're sure?

635
00:40:09,480 --> 00:40:10,410
You don't want the 92?

636
00:40:13,140 --> 00:40:15,641
How many people think
Drew is going to win?

637
00:40:15,641 --> 00:40:16,890
AUDIENCE: He's trying to lose.

638
00:40:16,890 --> 00:40:18,250
PROFESSOR: A couple people.

639
00:40:18,250 --> 00:40:20,909
Let's see.

640
00:40:20,909 --> 00:40:21,450
AUDIENCE: 91.

641
00:40:24,366 --> 00:40:26,320
It's either 91 or 93.

642
00:40:26,320 --> 00:40:29,870
PROFESSOR: 93, how did you
know there was a bigger number?

643
00:40:29,870 --> 00:40:31,229
Wow, very good.

644
00:40:31,229 --> 00:40:31,770
There you go.

645
00:40:31,770 --> 00:40:33,500
What was your strategy?

646
00:40:33,500 --> 00:40:36,580
AUDIENCE: I figured you'd
probably pick them within 1.

647
00:40:36,580 --> 00:40:38,068
PROFESSOR: Yeah, that's good.

648
00:40:38,068 --> 00:40:38,984
AUDIENCE: [INAUDIBLE].

649
00:40:49,972 --> 00:40:52,452
PROFESSOR: Actually, they're
both [INAUDIBLE] to 93.

650
00:40:52,452 --> 00:40:54,350
Would you still switch?

651
00:40:54,350 --> 00:40:56,560
AUDIENCE: [INAUDIBLE].

652
00:40:56,560 --> 00:40:57,600
PROFESSOR: Still switch.

653
00:40:57,600 --> 00:40:58,480
Hmm.

654
00:40:58,480 --> 00:41:00,518
All right, so is it 50-50 then?

655
00:41:00,518 --> 00:41:03,410
Because you see a 92 a 93,
you're switching either way.

656
00:41:03,410 --> 00:41:04,300
AUDIENCE: Yeah.

657
00:41:04,300 --> 00:41:05,635
PROFESSOR: 50-50.

658
00:41:05,635 --> 00:41:07,670
All right, how many
people still like 50-50?

659
00:41:07,670 --> 00:41:10,880
He can't beat 50-50 here.

660
00:41:10,880 --> 00:41:11,380
Any ideas?

661
00:41:11,380 --> 00:41:12,840
Can you beat 50-50?

662
00:41:12,840 --> 00:41:13,796
Yeah.

663
00:41:13,796 --> 00:41:16,664
AUDIENCE: I just pay
attention to your face.

664
00:41:16,664 --> 00:41:19,391
PROFESSOR: I guess
I gave it away.

665
00:41:19,391 --> 00:41:22,300
I think I've lost every bet
so far in the course here,

666
00:41:22,300 --> 00:41:24,320
even on Monty Hall, every time.

667
00:41:24,320 --> 00:41:27,450
So I guess I have a
tell or something here.

668
00:41:27,450 --> 00:41:31,210
Now, in fact, you
can beat 50-50.

669
00:41:31,210 --> 00:41:32,530
The information is helpful.

670
00:41:32,530 --> 00:41:34,835
Now, just to be clear,
are these numbers

671
00:41:34,835 --> 00:41:37,760
that I wrote down random?

672
00:41:37,760 --> 00:41:38,490
No.

673
00:41:38,490 --> 00:41:40,100
No, I'm trying
not to give away--

674
00:41:40,100 --> 00:41:42,180
my ice cream bill is going
through the roof here.

675
00:41:42,180 --> 00:41:43,320
I'm trying to make it hard.

676
00:41:43,320 --> 00:41:45,570
They're very much not random.

677
00:41:45,570 --> 00:41:48,400
There were two
smalls and two bigs.

678
00:41:48,400 --> 00:41:51,086
What is random here?

679
00:41:51,086 --> 00:41:52,460
AUDIENCE: Which
one he picked is.

680
00:41:52,460 --> 00:41:55,810
PROFESSOR: Which one he saw,
that's 50-50, effectively.

681
00:41:55,810 --> 00:41:58,920
So that's random there.

682
00:41:58,920 --> 00:42:03,810
Now his strategy could also
be random, but it wasn't.

683
00:42:03,810 --> 00:42:05,710
His strategy was, if
he sees a big number,

684
00:42:05,710 --> 00:42:07,320
he's swapping, which is odd.

685
00:42:07,320 --> 00:42:09,914
Most people see the big
number, they want to keep it.

686
00:42:09,914 --> 00:42:11,330
You know, I'd have
done well today

687
00:42:11,330 --> 00:42:13,621
if I had a really big number
and a really small number,

688
00:42:13,621 --> 00:42:15,470
because then I would
have won both times.

689
00:42:19,560 --> 00:42:23,530
OK, so to see why there might
be a winning strategy, or better

690
00:42:23,530 --> 00:42:27,510
than 50-50, imagine
that I had been nice

691
00:42:27,510 --> 00:42:30,020
and put a very small
number in one envelope

692
00:42:30,020 --> 00:42:32,640
and a very big number
of the other envelope.

693
00:42:32,640 --> 00:42:37,520
Say I had a 5 in one
and a 92 in the other.

694
00:42:37,520 --> 00:42:41,230
Can you win then,
if you know that?

695
00:42:41,230 --> 00:42:43,280
OK, how do you win if
you know there's one less

696
00:42:43,280 --> 00:42:46,100
than 10 and one bigger than 90?

697
00:42:46,100 --> 00:42:47,247
Yeah?

698
00:42:47,247 --> 00:42:49,330
AUDIENCE: If you get the
one less than 10, switch.

699
00:42:49,330 --> 00:42:50,705
PROFESSOR: Yeah,
and you're going

700
00:42:50,705 --> 00:42:52,820
to win with what probability?

701
00:42:52,820 --> 00:42:53,675
1.

702
00:42:53,675 --> 00:42:55,910
It's a certainty.

703
00:42:55,910 --> 00:43:03,700
All right, well, but
I'm not that nice, say.

704
00:43:03,700 --> 00:43:06,190
Say, though, there
is a threshold

705
00:43:06,190 --> 00:43:10,720
x, such that one of the
numbers is less than x,

706
00:43:10,720 --> 00:43:15,510
and one is bigger than
x, and you know x.

707
00:43:15,510 --> 00:43:16,330
Can you win now?

708
00:43:19,600 --> 00:43:22,750
Say you know that one number
is less than 47 and 1/2,

709
00:43:22,750 --> 00:43:25,700
and one is bigger
than 47 and 1/2.

710
00:43:25,700 --> 00:43:27,380
Can you win?

711
00:43:27,380 --> 00:43:31,030
Yeah, because if you get the
one less than 47 and 1/2,

712
00:43:31,030 --> 00:43:31,947
you switch.

713
00:43:31,947 --> 00:43:33,030
And otherwise you'll stay.

714
00:43:33,030 --> 00:43:36,240
So again, you win
with certainty.

715
00:43:36,240 --> 00:43:38,730
All right, that's good.

716
00:43:38,730 --> 00:43:40,130
Is there always such an x?

717
00:43:43,400 --> 00:43:46,810
You may not know it, but
is there always such an x?

718
00:43:46,810 --> 00:43:48,840
Yeah, because the
numbers are different.

719
00:43:48,840 --> 00:43:51,080
Just pick up a real
number in between them.

720
00:43:51,080 --> 00:43:54,110
You know, with 92 and
93, there's 92 and 1/2,

721
00:43:54,110 --> 00:43:55,670
is such an x.

722
00:43:55,670 --> 00:43:59,470
Now the only problem
is you don't know it.

723
00:43:59,470 --> 00:44:01,920
And there's no way to
figure it out that's

724
00:44:01,920 --> 00:44:04,580
within the rules of the game.

725
00:44:04,580 --> 00:44:08,970
Now in life, if you
don't know something,

726
00:44:08,970 --> 00:44:12,870
and you can't figure it
out, what can you do?

727
00:44:12,870 --> 00:44:13,670
AUDIENCE: Guess.

728
00:44:13,670 --> 00:44:15,340
PROFESSOR: Guess it.

729
00:44:15,340 --> 00:44:18,080
All right, now this turns out
to be a really good thing to do,

730
00:44:18,080 --> 00:44:21,757
and especially good in a lot
of computer science situations.

731
00:44:21,757 --> 00:44:23,590
If you don't know it,
and you can't know it,

732
00:44:23,590 --> 00:44:28,210
well you could try to guess
and you might be right.

733
00:44:28,210 --> 00:44:31,570
Now, if you guess x
and you are right,

734
00:44:31,570 --> 00:44:34,060
what's your
probability of winning?

735
00:44:34,060 --> 00:44:35,970
1.

736
00:44:35,970 --> 00:44:40,220
If you don't guess x, what's
your probability of winning?

737
00:44:40,220 --> 00:44:44,100
Say you guess wrong, but
you follow the rule that

738
00:44:44,100 --> 00:44:46,100
if it's less than what
you guessed, you swap it,

739
00:44:46,100 --> 00:44:47,560
otherwise you don't.

740
00:44:47,560 --> 00:44:51,370
What's your probability
of winning that?

741
00:44:51,370 --> 00:44:51,960
P?

742
00:44:51,960 --> 00:44:55,490
Well, yeah, what's P?

743
00:44:55,490 --> 00:44:58,840
It's a nice value of P.

744
00:44:58,840 --> 00:45:00,900
You know, say you
guessed-- well,

745
00:45:00,900 --> 00:45:04,719
say you weren't paying
attention and you guessed 200.

746
00:45:04,719 --> 00:45:07,010
And your rule is, if you're
less than what you guessed,

747
00:45:07,010 --> 00:45:09,650
you're less than 200,
you swap, which means

748
00:45:09,650 --> 00:45:12,040
you're just going to swap.

749
00:45:12,040 --> 00:45:14,690
And you started with a
random one, so what's

750
00:45:14,690 --> 00:45:17,370
your chance of winning?

751
00:45:17,370 --> 00:45:19,190
A half.

752
00:45:19,190 --> 00:45:21,670
So if you guessed wrong,
you didn't lose anything.

753
00:45:21,670 --> 00:45:23,580
You still win with
probability a half.

754
00:45:23,580 --> 00:45:27,860
If you guessed right, you
win with probability 1,

755
00:45:27,860 --> 00:45:29,610
and there's some chance
you guessed right,

756
00:45:29,610 --> 00:45:33,250
so now you've got a
strategy that beats 50-50,

757
00:45:33,250 --> 00:45:35,970
because you guessed.

758
00:45:35,970 --> 00:45:39,040
OK, and this is a whole
field in computer science

759
00:45:39,040 --> 00:45:41,840
where you get randomized
algorithms, where

760
00:45:41,840 --> 00:45:46,760
this strategy depends on a
coin flip, on guessing a value.

761
00:45:46,760 --> 00:45:51,360
And it leads to getting
potentially a better outcome.

762
00:45:51,360 --> 00:45:53,260
All right, so let's
prove that now

763
00:45:53,260 --> 00:45:57,320
and see what the probability
is of winning with the guess

764
00:45:57,320 --> 00:45:59,468
strategy and what's
a good way to guess.

765
00:46:07,290 --> 00:46:10,780
So we're going to first
formalize our random protocol.

766
00:46:21,350 --> 00:46:30,500
All right, so if this
is the winning strategy,

767
00:46:30,500 --> 00:46:42,020
well first the envelopes
contain y and z,

768
00:46:42,020 --> 00:46:45,730
and they're in the
interval 0 to n.

769
00:46:45,730 --> 00:46:47,587
And y is going to
be less than z.

770
00:46:47,587 --> 00:46:49,170
That's how we're
going to define them.

771
00:46:49,170 --> 00:46:53,290
Now you don't know
y and z, but y

772
00:46:53,290 --> 00:46:56,730
is the smaller number in the
envelope, z is the larger.

773
00:46:56,730 --> 00:47:01,230
And in our example, n was 100.

774
00:47:01,230 --> 00:47:12,540
Now the player
chooses x randomly,

775
00:47:12,540 --> 00:47:18,800
uniformly among all the
possible half integers.

776
00:47:21,520 --> 00:47:26,230
So he might pick 1/2, he might
pick 1 and 1/2, 2 and 1/2,

777
00:47:26,230 --> 00:47:29,844
all the way out to n minus 1/2.

778
00:47:29,844 --> 00:47:32,010
You know, because if the
player picks anything else,

779
00:47:32,010 --> 00:47:33,500
it's useless.

780
00:47:33,500 --> 00:47:35,590
He won't have a chance to win.

781
00:47:35,590 --> 00:47:37,880
But he might, in
the case of 5 and 6,

782
00:47:37,880 --> 00:47:40,570
might have picked 5 and 1/2.

783
00:47:40,570 --> 00:47:44,100
And we're going to make our
random guess be equally likely

784
00:47:44,100 --> 00:47:48,060
among those n values, because
well he doesn't really

785
00:47:48,060 --> 00:47:52,290
know what the numbers are
I put in the envelopes.

786
00:47:52,290 --> 00:47:56,976
If he sees 6, he doesn't
know if it's 5 or it's 7, OK?

787
00:47:56,976 --> 00:47:59,350
So that's why we're going to
pick them all equally likely

788
00:47:59,350 --> 00:48:00,058
and it's uniform.

789
00:48:02,860 --> 00:48:07,850
Now the player is hoping
that he picked a value that

790
00:48:07,850 --> 00:48:12,500
splits y and z, OK?

791
00:48:12,500 --> 00:48:15,329
Because then he's going to win.

792
00:48:15,329 --> 00:48:17,620
If he picked an x between
the numbers in the envelopes,

793
00:48:17,620 --> 00:48:21,860
and follows the swap, if he got
the small number less than x,

794
00:48:21,860 --> 00:48:27,050
he's going to win, all right?

795
00:48:27,050 --> 00:48:41,100
Then the player opens a
random envelope, 50-50,

796
00:48:41,100 --> 00:48:47,340
to reveal a number r,
which is either y or z,

797
00:48:47,340 --> 00:48:51,730
but the player doesn't
know which one.

798
00:48:51,730 --> 00:48:55,690
So one of the numbers
gets revealed.

799
00:48:55,690 --> 00:49:03,100
And then the last step
is the player swaps

800
00:49:03,100 --> 00:49:08,720
if the number he revealed is
less than the guessed split

801
00:49:08,720 --> 00:49:09,220
number.

802
00:49:12,580 --> 00:49:16,400
That's the strategy,
and it's a strategy

803
00:49:16,400 --> 00:49:22,720
that depends on a random
guess or a random number.

804
00:49:26,302 --> 00:49:27,760
So let's figure
out the probability

805
00:49:27,760 --> 00:49:30,060
of winning with that strategy.

806
00:49:30,060 --> 00:49:33,640
And we're going to
use the tree method.

807
00:49:33,640 --> 00:49:37,410
Well, the first branch
in the tree method

808
00:49:37,410 --> 00:49:40,700
is whether or not--
where the guess wound up.

809
00:49:40,700 --> 00:49:42,770
And there's three cases
for how the guess did.

810
00:49:45,320 --> 00:49:48,660
You might have guessed
too low, in which case,

811
00:49:48,660 --> 00:49:53,590
x is less than y,
and y is less than z.

812
00:49:53,590 --> 00:49:57,130
You might have guessed
perfectly, in which case

813
00:49:57,130 --> 00:50:02,570
x is between y and z,
equals not possible.

814
00:50:02,570 --> 00:50:04,840
Or you might have
guessed too high,

815
00:50:04,840 --> 00:50:08,470
in which case y is less
than z is less than x.

816
00:50:08,470 --> 00:50:13,545
All right, so here's low,
here's high, and here's

817
00:50:13,545 --> 00:50:17,270
an OK guess, a good guess.

818
00:50:17,270 --> 00:50:21,920
Now what's the probability
you guessed low, that you

819
00:50:21,920 --> 00:50:25,130
picked an x less than y?

820
00:50:25,130 --> 00:50:30,140
y is integers, x is
the half integers.

821
00:50:30,140 --> 00:50:33,690
What's the probability
to assign to that chance,

822
00:50:33,690 --> 00:50:36,925
that you guessed low?

823
00:50:36,925 --> 00:50:39,547
You could use the value
of y in your answer.

824
00:50:39,547 --> 00:50:41,630
You don't know y, but we
can write it on the tree.

825
00:50:44,790 --> 00:50:47,686
What if y was 1?

826
00:50:47,686 --> 00:50:49,185
The smallest number
in the envelopes

827
00:50:49,185 --> 00:50:54,310
is one, what's the probability
you guessed below 1?

828
00:50:57,660 --> 00:51:02,950
Not quite 0, one of your
possible end guesses

829
00:51:02,950 --> 00:51:05,920
would be too low,
namely, a half.

830
00:51:05,920 --> 00:51:10,170
If the smallest number
y were 2, what's

831
00:51:10,170 --> 00:51:13,340
the probability
you guess below 2?

832
00:51:13,340 --> 00:51:15,070
2/n.

833
00:51:15,070 --> 00:51:19,510
And in general, if y
is the smallest value,

834
00:51:19,510 --> 00:51:21,440
you've got y possible
guesses that it

835
00:51:21,440 --> 00:51:27,070
would be too low, namely all
the half integers less than y.

836
00:51:27,070 --> 00:51:31,920
So the probability
you guess low is y/n,

837
00:51:31,920 --> 00:51:34,190
because each one
is 1 in n chance.

838
00:51:34,190 --> 00:51:37,250
And there's y that are too low.

839
00:51:37,250 --> 00:51:39,070
What's the probability
your guess is good,

840
00:51:39,070 --> 00:51:40,430
that you split y and z?

841
00:51:44,930 --> 00:51:52,310
z minus y over n, because
they're-- between these

842
00:51:52,310 --> 00:51:56,910
integers, z and y, there's
z minus y half integers.

843
00:51:56,910 --> 00:51:59,792
Each could have been guessed
with probability 1/n.

844
00:51:59,792 --> 00:52:01,625
And what's the probability
you guessed high?

845
00:52:04,330 --> 00:52:07,560
n minus z over n,
because there's

846
00:52:07,560 --> 00:52:11,856
n minus g half
integers between z

847
00:52:11,856 --> 00:52:16,570
and n being the last possible
one, and minus the half there.

848
00:52:16,570 --> 00:52:19,620
All right, so we've
got the first branch.

849
00:52:19,620 --> 00:52:24,620
Now the next branch
is the revealed value.

850
00:52:24,620 --> 00:52:29,190
Did you open the smaller
one or the larger one?

851
00:52:29,190 --> 00:52:31,430
r equals y means you
opened the smaller one.

852
00:52:31,430 --> 00:52:33,300
r equals z means you
saw the bigger one.

853
00:52:42,580 --> 00:52:44,995
All right, if I've gone
down this branch where

854
00:52:44,995 --> 00:52:47,750
I guessed too low, but
I don't know that, yet,

855
00:52:47,750 --> 00:52:49,880
but say I've guessed
too low, what's

856
00:52:49,880 --> 00:52:52,120
the probability I opened
the smaller envelope?

857
00:52:55,830 --> 00:52:58,500
One half, because you're just
picking a random envelope

858
00:52:58,500 --> 00:52:59,870
and opening up.

859
00:52:59,870 --> 00:53:02,790
So these each happen
with probability a half.

860
00:53:02,790 --> 00:53:05,066
And that's true no matter what.

861
00:53:05,066 --> 00:53:05,940
They're all one half.

862
00:53:08,530 --> 00:53:13,370
All right, well, now we
can compute the probability

863
00:53:13,370 --> 00:53:14,910
of each outcome.

864
00:53:14,910 --> 00:53:19,490
This is y/n times
1/2, is y over 2n.

865
00:53:19,490 --> 00:53:22,920
This is y over 2n also.

866
00:53:22,920 --> 00:53:24,880
This is z minus y over 2n.

867
00:53:27,742 --> 00:53:31,090
z minus y over 2n.

868
00:53:31,090 --> 00:53:32,910
n minus z over 2n.

869
00:53:39,298 --> 00:53:42,030
All right, I got all
the sample points.

870
00:53:42,030 --> 00:53:45,530
I got all the probabilities.

871
00:53:45,530 --> 00:53:46,872
Let's figure out if you win now.

872
00:53:46,872 --> 00:53:49,080
And to do that, the first
thing we have to figure out

873
00:53:49,080 --> 00:53:49,995
is, do you swap?

874
00:53:52,650 --> 00:53:54,073
Do you swap here?

875
00:53:58,627 --> 00:53:59,460
Well, what happened?

876
00:53:59,460 --> 00:54:04,240
I revealed y, which is
bigger than my split value

877
00:54:04,240 --> 00:54:06,270
x, my guessed value.

878
00:54:06,270 --> 00:54:10,890
And so I am guessing I've
got the biggest value.

879
00:54:10,890 --> 00:54:13,800
So I don't swap.

880
00:54:13,800 --> 00:54:14,850
So there's no swap.

881
00:54:18,340 --> 00:54:19,060
What about here?

882
00:54:19,060 --> 00:54:20,140
Do I swap here?

883
00:54:23,070 --> 00:54:26,060
I open z, I saw z.

884
00:54:26,060 --> 00:54:29,890
z is bigger than what I
think the midpoint is,

885
00:54:29,890 --> 00:54:30,740
so I wouldn't swap.

886
00:54:30,740 --> 00:54:33,830
I only guess if
the value I reveal

887
00:54:33,830 --> 00:54:37,142
is less than my
guessed midpoint.

888
00:54:37,142 --> 00:54:40,270
If I got a value I see
bigger than my midpoint,

889
00:54:40,270 --> 00:54:42,256
I think I've got the big one.

890
00:54:42,256 --> 00:54:45,794
So there's no swap.

891
00:54:45,794 --> 00:54:46,460
What about here?

892
00:54:46,460 --> 00:54:48,490
Do I swap here?

893
00:54:48,490 --> 00:54:50,360
Yes, here I swap
because I open up

894
00:54:50,360 --> 00:54:52,714
something that's less
than my guessed midpoint,

895
00:54:52,714 --> 00:54:55,005
so I think I've got the small
one so I'm going to swap.

896
00:54:59,060 --> 00:54:59,750
What about here?

897
00:54:59,750 --> 00:55:01,620
Do I swap here?

898
00:55:01,620 --> 00:55:03,340
No swap, because I
opened up something

899
00:55:03,340 --> 00:55:05,070
that's bigger than midpoint.

900
00:55:05,070 --> 00:55:07,080
What about here?

901
00:55:07,080 --> 00:55:09,760
I swap.

902
00:55:09,760 --> 00:55:11,540
And what about here?

903
00:55:11,540 --> 00:55:13,532
Swap on both of them,
because both of them

904
00:55:13,532 --> 00:55:17,370
are smaller than my
guessed midpoint.

905
00:55:17,370 --> 00:55:21,210
All right, now let's figure
out if you won or you lost.

906
00:55:21,210 --> 00:55:26,522
So here I did not swap and I
started with a smaller value.

907
00:55:26,522 --> 00:55:27,230
So what happened?

908
00:55:27,230 --> 00:55:29,715
Did I win or lose?

909
00:55:29,715 --> 00:55:30,850
Lose.

910
00:55:30,850 --> 00:55:33,550
I opened he smaller
value and did not swap.

911
00:55:33,550 --> 00:55:35,970
What happened here?

912
00:55:35,970 --> 00:55:36,650
I win.

913
00:55:36,650 --> 00:55:40,020
I open the bigger
value and did not swap.

914
00:55:40,020 --> 00:55:42,350
Here what happened?

915
00:55:42,350 --> 00:55:45,950
Win, I opened the small value
when I swapped, that's a win.

916
00:55:45,950 --> 00:55:48,330
Here?

917
00:55:48,330 --> 00:55:51,280
I opened the big
value, did not swap.

918
00:55:51,280 --> 00:55:53,590
It's a win.

919
00:55:53,590 --> 00:55:57,590
Here I opened a small value
and swap, that's a win.

920
00:55:57,590 --> 00:56:02,812
And here I opened a big value
and swap, that's a lose.

921
00:56:02,812 --> 00:56:05,610
All right, now we can compute
the probability of winning.

922
00:56:08,240 --> 00:56:12,540
The probability of a win is
the sum of these four sample

923
00:56:12,540 --> 00:56:22,110
points-- y over 2n plus
z minus y over 2n plus

924
00:56:22,110 --> 00:56:29,220
z minus y-- whoops-- over
2n plus n minus z over 2n.

925
00:56:33,090 --> 00:56:38,830
That equals-- z cancels the
z, and a y cancels the y,

926
00:56:38,830 --> 00:56:44,820
so I've got n, one left,
n left, one z left,

927
00:56:44,820 --> 00:56:49,730
and one negative y left over 2n.

928
00:56:49,730 --> 00:56:50,990
And I can simplify that.

929
00:56:50,990 --> 00:56:55,420
N over 2n is 1/2, plus
now what's left over

930
00:56:55,420 --> 00:56:59,430
is z minus y over 2n.

931
00:57:02,140 --> 00:57:06,960
And we know the z and y are
different by at least 1.

932
00:57:06,960 --> 00:57:11,270
So this is at least
1/2 plus 1 over 2n.

933
00:57:14,360 --> 00:57:20,280
And so if n is 100, you've got
a 50 and 1/2% chance of winning.

934
00:57:20,280 --> 00:57:24,930
If n is 10, the numbers
are from 0 to 10,

935
00:57:24,930 --> 00:57:28,220
you've got at least
a 55% chance to win,

936
00:57:28,220 --> 00:57:31,430
which is pretty high, OK?

937
00:57:33,990 --> 00:57:35,700
Any questions here?

938
00:57:35,700 --> 00:57:36,700
You see what's going on?

939
00:57:36,700 --> 00:57:39,660
So here's the zone
where you guessed right.

940
00:57:39,660 --> 00:57:41,190
You get the win either way.

941
00:57:41,190 --> 00:57:43,630
Here you guess low, and it
doesn't make any difference.

942
00:57:43,630 --> 00:57:44,940
You're 50-50.

943
00:57:44,940 --> 00:57:48,690
You guessed high and
you're 50-50 again.

944
00:57:48,690 --> 00:57:52,920
So guessing helps.

945
00:57:52,920 --> 00:57:54,500
Any questions about that?

946
00:57:54,500 --> 00:57:55,532
Yeah.

947
00:57:55,532 --> 00:57:58,484
AUDIENCE: So aren't you
assuming that-- because you say,

948
00:57:58,484 --> 00:58:01,436
have a range, a greater
range of numbers,

949
00:58:01,436 --> 00:58:04,224
that's there's a greater
chance that the number is

950
00:58:04,224 --> 00:58:07,840
going fall in that range.

951
00:58:07,840 --> 00:58:10,050
PROFESSOR: I was
the nasty guy here.

952
00:58:10,050 --> 00:58:12,720
I picked z and y to be
consecutive numbers, which

953
00:58:12,720 --> 00:58:16,575
minimized your chance of
doing well with this strategy.

954
00:58:16,575 --> 00:58:18,449
AUDIENCE: Because it
seems like we don't know

955
00:58:18,449 --> 00:58:19,574
anything about [INAUDIBLE].

956
00:58:22,746 --> 00:58:24,245
PROFESSOR: The
distribution of what?

957
00:58:24,245 --> 00:58:26,579
AUDIENCE: Of possible
other numbers

958
00:58:26,579 --> 00:58:27,620
in envelopes [INAUDIBLE].

959
00:58:27,620 --> 00:58:30,000
PROFESSOR: Right, you know
nothing about the distribution

960
00:58:30,000 --> 00:58:32,120
because there is none.

961
00:58:32,120 --> 00:58:33,530
There's no randomness there.

962
00:58:33,530 --> 00:58:36,900
I picked worst
case numbers here.

963
00:58:36,900 --> 00:58:41,700
AUDIENCE: But it's still
sort of like a [INAUDIBLE].

964
00:58:41,700 --> 00:58:43,620
If it's 10, you still
have a better chance

965
00:58:43,620 --> 00:58:48,120
to swap than if you
[INAUDIBLE] below that.

966
00:58:48,120 --> 00:58:49,460
PROFESSOR: No.

967
00:58:49,460 --> 00:58:52,320
No, any deterministic
strategy you pick,

968
00:58:52,320 --> 00:58:54,470
if you don't guess
that random x,

969
00:58:54,470 --> 00:58:58,110
you will not do
better than 50-50.

970
00:58:58,110 --> 00:59:01,470
The only way you'd beat
50-50 is to, in your mind,

971
00:59:01,470 --> 00:59:07,390
make a random number 13
and 1/2 and flip if you

972
00:59:07,390 --> 00:59:09,780
see less than 13 and 1/2.

973
00:59:09,780 --> 00:59:11,640
If you come in with
a strategy that hey,

974
00:59:11,640 --> 00:59:15,350
10 is a small
number, out of 100,

975
00:59:15,350 --> 00:59:19,380
and so I'm going to
swap if I see a 10,

976
00:59:19,380 --> 00:59:22,600
well, no, because first it
could be-- I might have done

977
00:59:22,600 --> 00:59:23,720
9 or 11, either one.

978
00:59:23,720 --> 00:59:26,470
In fact, if I know
that's your strategy,

979
00:59:26,470 --> 00:59:29,530
I'm going to make a 9.

980
00:59:29,530 --> 00:59:32,620
And then you're
doomed in seeing a 10.

981
00:59:32,620 --> 00:59:34,870
AUDIENCE: So is there a
best way to pick that?

982
00:59:34,870 --> 00:59:38,680
PROFESSOR: Yes, I also
have an optimal strategy.

983
00:59:38,680 --> 00:59:41,350
In fact, there's two
interesting things

984
00:59:41,350 --> 00:59:43,630
about this game
which we won't prove.

985
00:59:43,630 --> 00:59:46,500
The first thing we'll prove,
we will prove but it's true,

986
00:59:46,500 --> 00:59:50,470
is this is the optimal
strategy for you.

987
00:59:50,470 --> 00:59:53,520
And my optimal
strategy is to pick

988
00:59:53,520 --> 00:59:57,820
a random value of y
between 0 and n minus 1,

989
00:59:57,820 --> 01:00:01,217
and then to make z be one more.

990
01:00:01,217 --> 01:00:03,050
And the way to think
about this, and there's

991
01:00:03,050 --> 01:00:06,520
whole classes you can
take on game theory,

992
01:00:06,520 --> 01:00:11,170
is that suppose I do
pick my guys randomly

993
01:00:11,170 --> 01:00:14,900
by picking a random y
uniformly from 0 to n minus 1.

994
01:00:14,900 --> 01:00:18,510
So my smaller number is
anything between 0 and 99,

995
01:00:18,510 --> 01:00:22,550
equally likely, and then the
bigger number's just one more.

996
01:00:22,550 --> 01:00:24,340
So if I picked 92
for the random one,

997
01:00:24,340 --> 01:00:26,360
the next one's going to be 93.

998
01:00:26,360 --> 01:00:30,350
If that's my strategy,
even if I tell you that,

999
01:00:30,350 --> 01:00:35,354
you cannot do any better than
getting 1/2 plus 1 over 2n

1000
01:00:35,354 --> 01:00:37,770
probability of winning, no
matter what you do in the whole

1001
01:00:37,770 --> 01:00:40,420
world, OK?

1002
01:00:40,420 --> 01:00:43,620
And if you use
this strategy here,

1003
01:00:43,620 --> 01:00:47,950
no matter what I do, even
knowing that's your strategy,

1004
01:00:47,950 --> 01:00:52,730
I can't keep you from
getting this much.

1005
01:00:52,730 --> 01:00:55,620
It's called a minimax solution.

1006
01:00:55,620 --> 01:00:59,170
So my optimal strategy is to
pick uniformly in 0 to 99,

1007
01:00:59,170 --> 01:01:00,470
and the next one's one more.

1008
01:01:00,470 --> 01:01:02,880
Your optimal strategy is this.

1009
01:01:02,880 --> 01:01:06,447
Pick a number uniformly in
the half integers there,

1010
01:01:06,447 --> 01:01:08,030
and swap if you see
something smaller.

1011
01:01:08,030 --> 01:01:10,310
And there's no better strategy.

1012
01:01:10,310 --> 01:01:12,270
In fact, any
deterministic strategy,

1013
01:01:12,270 --> 01:01:14,220
you don't do better than 50-50.

1014
01:01:14,220 --> 01:01:20,640
The optimal strategies require
randomness for each of us.

1015
01:01:20,640 --> 01:01:27,320
OK, any more questions about
what happened here, this game?

1016
01:01:27,320 --> 01:01:28,642
Yeah.

1017
01:01:28,642 --> 01:01:30,205
AUDIENCE: [INAUDIBLE]
when you see

1018
01:01:30,205 --> 01:01:32,971
a small number you're going
to swap, does that just mean

1019
01:01:32,971 --> 01:01:35,380
[INAUDIBLE]?

1020
01:01:35,380 --> 01:01:39,220
PROFESSOR: Yeah, basically,
yeah, that's what it means.

1021
01:01:39,220 --> 01:01:41,650
Effectively, that 50
is your random number.

1022
01:01:41,650 --> 01:01:44,180
If you think OK, if I see less
than 50, I'm going to swap,

1023
01:01:44,180 --> 01:01:45,240
bigger than 50 I don't.

1024
01:01:45,240 --> 01:01:46,710
Now you have to
decide what happens

1025
01:01:46,710 --> 01:01:49,010
at 50, if you saw exactly 50.

1026
01:01:49,010 --> 01:01:52,819
So you'd probably go in at
49 and 1/2 and 50 and 1/2.

1027
01:01:52,819 --> 01:01:54,610
And so that could be
construed as that way.

1028
01:01:54,610 --> 01:01:57,400
But you didn't pick it
randomly, and that sort

1029
01:01:57,400 --> 01:02:00,130
of human intuition,
to swap so I could

1030
01:02:00,130 --> 01:02:02,980
use that now to
design my strategy,

1031
01:02:02,980 --> 01:02:07,780
knowing that your random
numbers could be 50 and 1/2, OK?

1032
01:02:10,500 --> 01:02:12,010
No, I'll be 50-50
because what I'll

1033
01:02:12,010 --> 01:02:13,740
do, if I know that's
your strategy,

1034
01:02:13,740 --> 01:02:14,990
my numbers will be 1 and 2.

1035
01:02:17,760 --> 01:02:19,480
Well, when you do
these analyses,

1036
01:02:19,480 --> 01:02:23,540
you sort of, if it's declared
that's it, I can assume that.

1037
01:02:23,540 --> 01:02:25,486
And I might actually guess that.

1038
01:02:25,486 --> 01:02:28,150
In fact, that's why I did pick
two very small numbers and two

1039
01:02:28,150 --> 01:02:29,590
very big numbers.

1040
01:02:29,590 --> 01:02:31,170
But given the
perverse nature here,

1041
01:02:31,170 --> 01:02:33,830
when you saw the big
number, you got rid of it,

1042
01:02:33,830 --> 01:02:37,780
and when you saw the
small number, you kept it.

1043
01:02:37,780 --> 01:02:39,820
So you sort of did
reverse psychology

1044
01:02:39,820 --> 01:02:43,900
and it worked out, by chance,
that it worked out that.

1045
01:02:43,900 --> 01:02:47,047
Normally it would go
the other way, I think.

1046
01:02:47,047 --> 01:02:48,880
Of course, you've all
seen enough games now,

1047
01:02:48,880 --> 01:02:53,800
you know to do the
non-standard thing, I think.

1048
01:02:53,800 --> 01:02:56,409
Now, this kind of thinking
comes up all the time

1049
01:02:56,409 --> 01:02:57,700
in computer science algorithms.

1050
01:02:57,700 --> 01:02:59,550
You know, the
protocol that's used

1051
01:02:59,550 --> 01:03:02,720
to communicate across
a network, ethernet,

1052
01:03:02,720 --> 01:03:05,750
shared bus is randomized.

1053
01:03:05,750 --> 01:03:09,060
Each entity that wants to use
the bus flips a random number,

1054
01:03:09,060 --> 01:03:10,040
flips a random coin.

1055
01:03:10,040 --> 01:03:11,430
You could think of it that way.

1056
01:03:11,430 --> 01:03:14,270
And it broadcasts
with that probability.

1057
01:03:14,270 --> 01:03:16,150
And if there's a
collision, it backs off

1058
01:03:16,150 --> 01:03:18,530
and chooses a smaller
probability next time.

1059
01:03:18,530 --> 01:03:21,400
And if it gets through, then it
tries to cram a bunch of stuff

1060
01:03:21,400 --> 01:03:24,060
through, until you
get a collision again.

1061
01:03:24,060 --> 01:03:26,680
The best algorithms for
sorting a list of numbers

1062
01:03:26,680 --> 01:03:29,780
are randomized, quick sort,
which you'll see in 6046,

1063
01:03:29,780 --> 01:03:31,800
is a randomized algorithm.

1064
01:03:31,800 --> 01:03:33,170
In fact, it's not dissimilar.

1065
01:03:33,170 --> 01:03:37,017
You guess a random value
and split all the numbers

1066
01:03:37,017 --> 01:03:38,100
based on the random value.

1067
01:03:38,100 --> 01:03:41,030
Who's bigger, who's smaller,
and then you [INAUDIBLE].

1068
01:03:41,030 --> 01:03:44,670
You do things like that.

1069
01:03:44,670 --> 01:03:49,900
OK, that's all I'm going to say
about uniform distributions.

1070
01:03:49,900 --> 01:03:54,400
Next I want to talk about
the binomial distribution.

1071
01:03:54,400 --> 01:03:59,300
And this is the most
important distribution

1072
01:03:59,300 --> 01:04:02,510
in computer science, and
probably the most important

1073
01:04:02,510 --> 01:04:07,730
in all of discrete
calculations in the sciences.

1074
01:04:07,730 --> 01:04:10,480
In continuous distributions,
you get the normal distribution.

1075
01:04:10,480 --> 01:04:16,562
But, for discrete problems, the
binomial is the most important.

1076
01:04:16,562 --> 01:04:17,910
And there's two versions.

1077
01:04:17,910 --> 01:04:21,820
There's the unbiased
binomial distribution

1078
01:04:21,820 --> 01:04:23,301
and then there's the biased one.

1079
01:04:28,462 --> 01:04:30,170
Now this one's a little
more complicated.

1080
01:04:30,170 --> 01:04:34,460
You've got a parameter n again,
and the distribution function

1081
01:04:34,460 --> 01:04:37,130
on k, probability
of getting k is

1082
01:04:37,130 --> 01:04:42,000
n choose k times
2 to the minus n.

1083
01:04:42,000 --> 01:04:48,560
And n is at least 1, and
k is between 0 and n.

1084
01:04:48,560 --> 01:05:01,790
And then for the general
binomial distribution,

1085
01:05:01,790 --> 01:05:04,140
we have fn of k, the
probability of getting

1086
01:05:04,140 --> 01:05:10,270
k is n choose k times--
actually, it's fnp,

1087
01:05:10,270 --> 01:05:12,500
there's another
parameter p here.

1088
01:05:12,500 --> 01:05:17,585
p to the k times 1 minus
p to the n minus k power.

1089
01:05:17,585 --> 01:05:20,640
And p is a value between
0 to 1 typically.

1090
01:05:20,640 --> 01:05:23,330
It's a probability of
something happening.

1091
01:05:23,330 --> 01:05:26,560
So in the general
case, you've got nnp.

1092
01:05:26,560 --> 01:05:30,750
The unbiased case
corresponds to when p is 1/2.

1093
01:05:30,750 --> 01:05:35,200
Because if p is 1/2, you get 2
to the minus k and 2 the minus

1094
01:05:35,200 --> 01:05:38,600
n minus k, and that's
just 2 to the minus n.

1095
01:05:38,600 --> 01:05:40,990
So you can think of this
as the case when p is 1/2.

1096
01:05:46,330 --> 01:05:50,100
All right, to give
you an example why

1097
01:05:50,100 --> 01:05:54,230
this is so important, why
it comes up all the time.

1098
01:05:54,230 --> 01:05:59,030
Imagine that you've got a
system with n components.

1099
01:05:59,030 --> 01:06:03,159
And that each component
fails with probability p.

1100
01:06:03,159 --> 01:06:05,590
And you want to know the
probability of having

1101
01:06:05,590 --> 01:06:07,565
some number of failures.

1102
01:06:15,160 --> 01:06:22,890
So for example,
there's n components

1103
01:06:22,890 --> 01:06:32,900
and each fails independently--
in fact, mutually

1104
01:06:32,900 --> 01:06:40,190
independently--
with probability p.

1105
01:06:42,790 --> 01:06:44,620
And p will be between 0 and 1.

1106
01:06:47,450 --> 01:06:51,400
And now we're interested
in the number of failures,

1107
01:06:51,400 --> 01:06:54,250
so we'll make R be a
random variable that

1108
01:06:54,250 --> 01:06:57,245
tells us the number
of failed components.

1109
01:07:02,810 --> 01:07:05,750
And it turns out
the answer is simply

1110
01:07:05,750 --> 01:07:09,420
that function, which
we'll prove is a theorem.

1111
01:07:09,420 --> 01:07:19,005
The probability that R equals k
is simply fnp of k [INAUDIBLE].

1112
01:07:23,870 --> 01:07:26,310
So the general
binomial distribution

1113
01:07:26,310 --> 01:07:29,850
gives you the distribution
on the number of failures

1114
01:07:29,850 --> 01:07:32,060
in a system with
n components where

1115
01:07:32,060 --> 01:07:34,638
they fail with probability p.

1116
01:07:34,638 --> 01:07:36,120
So let's prove that.

1117
01:07:47,990 --> 01:07:51,920
To do that, we're going to
construct our tree again.

1118
01:07:51,920 --> 01:07:55,000
It's a little big because
you've got n components.

1119
01:07:55,000 --> 01:07:57,180
But you look at the
first component,

1120
01:07:57,180 --> 01:08:00,480
and that can fail or not.

1121
01:08:00,480 --> 01:08:02,430
And it fails a probability p.

1122
01:08:02,430 --> 01:08:05,710
It's OK with
probability 1 minus p.

1123
01:08:05,710 --> 01:08:08,132
And you have the
second component.

1124
01:08:08,132 --> 01:08:10,740
It can fail or not.

1125
01:08:10,740 --> 01:08:12,390
Again p and 1 minus p.

1126
01:08:20,210 --> 01:08:22,733
And you keep on going until
you get to the nth component.

1127
01:08:27,229 --> 01:08:32,850
And that can fail or not, the
probability p or 1 minus p,

1128
01:08:32,850 --> 01:08:33,950
in general down here.

1129
01:08:40,140 --> 01:08:42,960
And now we look at all the
sample points out here.

1130
01:08:47,120 --> 01:08:54,160
And it's all length and
vectors of failure or not.

1131
01:08:54,160 --> 01:08:58,979
So the top sample point here
would be n good components.

1132
01:09:01,750 --> 01:09:05,370
All right, the next one would
be the first n minus 1 are good,

1133
01:09:05,370 --> 01:09:08,420
the last one failed.

1134
01:09:08,420 --> 01:09:10,170
All the way down
to the very bottom,

1135
01:09:10,170 --> 01:09:11,380
you can have all n fail.

1136
01:09:14,871 --> 01:09:17,370
All right, so how many sample
points are there in the sample

1137
01:09:17,370 --> 01:09:23,170
space with n components?

1138
01:09:23,170 --> 01:09:24,830
2 to the n, all right?

1139
01:09:24,830 --> 01:09:26,760
Because you've got n positions.

1140
01:09:26,760 --> 01:09:28,960
There's two choices
for each value there.

1141
01:09:32,160 --> 01:09:35,010
Now how many of
the sample points

1142
01:09:35,010 --> 01:09:40,109
have exactly k
failed components?

1143
01:09:40,109 --> 01:09:43,210
Out of the 2 to the
n, how many correspond

1144
01:09:43,210 --> 01:09:44,819
to k of the components fail?

1145
01:09:48,779 --> 01:09:51,820
n choose k-- now this
goes back to counting.

1146
01:09:51,820 --> 01:09:55,690
Remember the
binomial coefficient.

1147
01:09:55,690 --> 01:10:06,960
So there are n choose
k sample points have

1148
01:10:06,960 --> 01:10:08,400
k failed components.

1149
01:10:16,350 --> 01:10:18,770
All right, what's the
probability of a sample

1150
01:10:18,770 --> 01:10:21,560
point with k failed components?

1151
01:10:24,184 --> 01:10:30,950
Well, I took k branches
with a failure.

1152
01:10:30,950 --> 01:10:36,480
Each of those gives
me a p, p to the k.

1153
01:10:36,480 --> 01:10:40,900
And I took n minus k
branches with no failure.

1154
01:10:40,900 --> 01:10:43,340
Each one of those
multiplies by a 1 minus p.

1155
01:10:46,370 --> 01:10:50,080
So no matter how I
got my k failures,

1156
01:10:50,080 --> 01:10:54,000
the probability for a particular
sample point with k failures

1157
01:10:54,000 --> 01:10:58,170
is just that, because I had
p get factored in k times,

1158
01:10:58,170 --> 01:11:00,900
in the failures, 1
minus p get factored

1159
01:11:00,900 --> 01:11:03,381
in n minus k times
for the situations

1160
01:11:03,381 --> 01:11:04,630
where it worked out all right.

1161
01:11:07,970 --> 01:11:10,980
So I've got this
many sample points.

1162
01:11:10,980 --> 01:11:22,870
Each of them has probability
p to the k times 1

1163
01:11:22,870 --> 01:11:25,300
minus p n minus k.

1164
01:11:28,230 --> 01:11:30,420
Any questions about that,
why that's the case?

1165
01:11:33,692 --> 01:11:35,650
All right, so now I can
compute the probability

1166
01:11:35,650 --> 01:11:38,730
there are k failures.

1167
01:11:38,730 --> 01:11:43,040
The probability that
R equals k is simply n

1168
01:11:43,040 --> 01:11:47,870
choose k, number of sample
points times their probability,

1169
01:11:47,870 --> 01:11:51,920
since they're all the
same for k failures.

1170
01:11:51,920 --> 01:11:55,830
And that, of course,
is just the formula

1171
01:11:55,830 --> 01:11:59,290
for the general
binomial distribution.

1172
01:11:59,290 --> 01:12:06,146
So that equals fnp of k, which
is what we are trying to prove.

1173
01:12:11,570 --> 01:12:14,050
OK.

1174
01:12:14,050 --> 01:12:16,840
Any questions about that?

1175
01:12:16,840 --> 01:12:18,980
So that's why it's
important, because there's

1176
01:12:18,980 --> 01:12:22,100
a lot of situations where
you're interested in the number

1177
01:12:22,100 --> 01:12:24,330
of-- if you had n
possible things,

1178
01:12:24,330 --> 01:12:26,590
what's the chance
k of them happen?

1179
01:12:26,590 --> 01:12:30,630
And it's just given
by this formula.

1180
01:12:30,630 --> 01:12:36,850
Now you can calculate this,
but, if I just told you,

1181
01:12:36,850 --> 01:12:45,310
for example, maybe I'm looking
at n is 100 and even p is 1/2,

1182
01:12:45,310 --> 01:12:47,490
you know, what's the
probability that we'd

1183
01:12:47,490 --> 01:12:51,690
get 25-- say, what's
the probably if getting

1184
01:12:51,690 --> 01:12:54,140
50 failed components?

1185
01:12:54,140 --> 01:12:58,480
Or if I flip 100 coins
and they're fair.

1186
01:12:58,480 --> 01:13:00,730
So probability half
of getting a heads.

1187
01:13:00,730 --> 01:13:05,680
What's the chance I actually
get 50 heads in 100 coins?

1188
01:13:05,680 --> 01:13:10,950
Looking at that, it's not
so clear what the answer is.

1189
01:13:10,950 --> 01:13:14,570
In fact, let's test
intuition here, for this.

1190
01:13:14,570 --> 01:13:16,140
We're going to
take a little vote

1191
01:13:16,140 --> 01:13:19,720
to see how good you are
looking at that formula,

1192
01:13:19,720 --> 01:13:23,420
or what your intuition is about
the probability of getting

1193
01:13:23,420 --> 01:13:24,695
exactly 50 heads.

1194
01:13:29,240 --> 01:13:34,534
It could be between 1 and
1/2, a half and a tenth,

1195
01:13:34,534 --> 01:13:40,030
a tenth and a hundredth, a
hundredth and one thousandth,

1196
01:13:40,030 --> 01:13:44,259
a thousandth and a
millionth, and 0.

1197
01:13:44,259 --> 01:13:46,050
I want to know what's
the probability, when

1198
01:13:46,050 --> 01:13:48,810
you flip 100 mutually
independent coins,

1199
01:13:48,810 --> 01:13:52,370
you get exactly 50 heads?

1200
01:13:52,370 --> 01:13:54,870
How many people think
it's at least a half,

1201
01:13:54,870 --> 01:13:57,900
that you get half heads?

1202
01:13:57,900 --> 01:13:58,920
Nobody likes that.

1203
01:13:58,920 --> 01:14:02,640
How many people think it's
between a half and a tenth?

1204
01:14:02,640 --> 01:14:03,890
One vote.

1205
01:14:03,890 --> 01:14:06,280
A tenth and a hundredth?

1206
01:14:06,280 --> 01:14:06,780
More.

1207
01:14:08,939 --> 01:14:10,730
Doesn't it have to be
at least a hundredth?

1208
01:14:10,730 --> 01:14:12,354
That's sort of the
most likely outcome.

1209
01:14:12,354 --> 01:14:16,600
There's only-- how many think
a hundredth and a thousandth?

1210
01:14:16,600 --> 01:14:17,350
All right.

1211
01:14:17,350 --> 01:14:19,840
A thousandth and a millionth?

1212
01:14:19,840 --> 01:14:22,040
Whoa, so you've got
to believe it is not

1213
01:14:22,040 --> 01:14:25,560
likely to get exactly 50
heads when I flip 100 coins.

1214
01:14:25,560 --> 01:14:28,690
All right, well the
answer is actually

1215
01:14:28,690 --> 01:14:30,660
between a tenth and
a hundredth here.

1216
01:14:33,730 --> 01:14:34,910
It's about 8%.

1217
01:14:34,910 --> 01:14:36,570
And we're going to compute that.

1218
01:14:36,570 --> 01:14:38,330
But here's another one.

1219
01:14:38,330 --> 01:14:40,370
I flip 100 coins.

1220
01:14:40,370 --> 01:14:46,520
What's the probability of
getting at most 25 heads?

1221
01:14:46,520 --> 01:14:48,605
So here, doesn't have
to be exactly 25.

1222
01:14:48,605 --> 01:14:53,400
It could be 25, 24, 23, 22,
all the way down to no heads.

1223
01:14:53,400 --> 01:14:55,560
So a lot of chances, all right?

1224
01:14:55,560 --> 01:14:57,630
To get at most 25 heads.

1225
01:14:57,630 --> 01:15:00,200
How many people think it's here?

1226
01:15:00,200 --> 01:15:02,770
Some, yeah, you've
got a lot of chances.

1227
01:15:02,770 --> 01:15:04,885
How about here?

1228
01:15:04,885 --> 01:15:06,230
Yeah, OK.

1229
01:15:06,230 --> 01:15:09,858
What about between a
tenth and a hundredth?

1230
01:15:09,858 --> 01:15:12,940
One person, two people left.

1231
01:15:12,940 --> 01:15:15,930
Between a hundredth
and a thousandth?

1232
01:15:15,930 --> 01:15:16,620
Nobody left?

1233
01:15:16,620 --> 01:15:18,050
Anybody here?

1234
01:15:18,050 --> 01:15:20,630
Thousandth and a millionth?

1235
01:15:20,630 --> 01:15:22,340
One person is left,
less than a million.

1236
01:15:22,340 --> 01:15:24,630
1 in a million?

1237
01:15:24,630 --> 01:15:25,880
There's the contrarian.

1238
01:15:25,880 --> 01:15:31,310
And you're right, the chance
of getting 25 or less heads

1239
01:15:31,310 --> 01:15:34,330
is less than 1 in a million.

1240
01:15:34,330 --> 01:15:39,020
So if somebody does it, his
name better be Persi Diaconis.

1241
01:15:39,020 --> 01:15:42,710
To get it consistently to 70,
[INAUDIBLE] 25 or fewer heads.

1242
01:15:42,710 --> 01:15:48,160
You get 75 tails or more is
extremely unlikely, all right?

1243
01:15:48,160 --> 01:15:50,660
So to see why that's
the case, we've

1244
01:15:50,660 --> 01:15:53,900
got to do a little
work on this formula.

1245
01:15:53,900 --> 01:15:57,820
Because we're saying that if
we sum this up for k ending 100

1246
01:15:57,820 --> 01:16:01,470
and k being 0 to 25,
and p being a half,

1247
01:16:01,470 --> 01:16:04,600
it's an incredibly tiny number.

1248
01:16:04,600 --> 01:16:05,922
So let's see why that is.

1249
01:16:05,922 --> 01:16:07,380
And this phenomenon
is course going

1250
01:16:07,380 --> 01:16:09,690
to be important in
computer science

1251
01:16:09,690 --> 01:16:12,290
because it's going to enable
you to tell almost exactly how

1252
01:16:12,290 --> 01:16:15,120
many components are going
to fail in a system,

1253
01:16:15,120 --> 01:16:16,617
if they're mutually independent.

1254
01:16:30,624 --> 01:16:33,040
All right, now I'm not going
to drag you through the math.

1255
01:16:33,040 --> 01:16:36,760
It's bad enough I'm going
to write it on the board.

1256
01:16:36,760 --> 01:16:40,320
There's a bunch of
this in the text.

1257
01:16:40,320 --> 01:16:43,560
And instead of k, I'm going to
represent by a parameter alpha

1258
01:16:43,560 --> 01:16:46,452
times n to be the integer k.

1259
01:16:46,452 --> 01:16:50,330
All right, this will replace
k with a new parameter alpha.

1260
01:16:50,330 --> 01:16:54,190
It's at most, and also it's
asymptotically equivalent

1261
01:16:54,190 --> 01:16:58,390
using tilde-- both of those are
true-- to this nasty looking

1262
01:16:58,390 --> 01:17:08,900
expression, 2 to the
alpha log p over alpha

1263
01:17:08,900 --> 01:17:18,740
plus 1 minus alpha log 1 minus
p over 1 minus alpha times

1264
01:17:18,740 --> 01:17:26,000
and n-- that's a big number,
it's 100-- over square root 2

1265
01:17:26,000 --> 01:17:31,840
pi alpha 1 minus alpha n.

1266
01:17:31,840 --> 01:17:33,875
And this, of course,
alpha is between 0 and 1.

1267
01:17:37,790 --> 01:17:39,970
Now when you do the
derivative on this thing,

1268
01:17:39,970 --> 01:17:42,840
it turns out its maximum
value is when alpha equals p.

1269
01:17:51,320 --> 01:17:57,580
And when you have alpha equals
p, you get log of 1 is 0,

1270
01:17:57,580 --> 01:17:58,640
log of 1 is 0.

1271
01:17:58,640 --> 01:18:03,090
All this messy stuff goes
away and we just get that.

1272
01:18:03,090 --> 01:18:10,520
And so in that case, fnp
of pn is at most, and also

1273
01:18:10,520 --> 01:18:19,130
asymptotically equal to 1 over
square root 2 pi p 1 minus p n.

1274
01:18:21,650 --> 01:18:23,860
And now you can plug in
values and compute things.

1275
01:18:23,860 --> 01:18:26,880
For example, if
I flip 100 coins,

1276
01:18:26,880 --> 01:18:29,720
and I want to know the
probability of getting 50--

1277
01:18:29,720 --> 01:18:32,240
that means alpha's a
half, and they're fair,

1278
01:18:32,240 --> 01:18:38,080
which means p is a
half, the answer is 8%.

1279
01:18:38,080 --> 01:18:42,240
So n equals 100 coins,
p is a half, then

1280
01:18:42,240 --> 01:18:44,730
the probability of 50 heads.

1281
01:18:49,370 --> 01:18:55,840
I just plug in p is a half here,
I get 1 over square root 50 pi,

1282
01:18:55,840 --> 01:19:00,670
which equals 0.080 and so forth.

1283
01:19:00,670 --> 01:19:05,905
So there's an 8% chance of
getting exactly 50 heads.

1284
01:19:05,905 --> 01:19:07,280
Now let's look at
the probability

1285
01:19:07,280 --> 01:19:10,670
of getting 25 or fewer.

1286
01:19:10,670 --> 01:19:13,548
And we'll see why that
is so surprisingly small.

1287
01:19:30,170 --> 01:19:38,160
OK, so for n equal
100 and p equal

1288
01:19:38,160 --> 01:19:42,630
a half an alpha equal a quarter,
because we want 25 heads,

1289
01:19:42,630 --> 01:19:46,110
I'll get exactly 25 heads first.

1290
01:19:46,110 --> 01:19:52,460
Probability of 25
heads is at most,

1291
01:19:52,460 --> 01:19:57,900
well that square root thing
comes out to about 0.09.

1292
01:19:57,900 --> 01:20:00,230
Then I get 2 to something.

1293
01:20:00,230 --> 01:20:09,380
I get 2 to the minus, all those
logs come out the 0.1887 times

1294
01:20:09,380 --> 01:20:12,920
n, which is the
kicker, that's 100.

1295
01:20:12,920 --> 01:20:16,730
So I get a 2 to the
minus 18th here.

1296
01:20:16,730 --> 01:20:21,080
And that makes this
thing be smaller than

1297
01:20:21,080 --> 01:20:26,230
or equal about to 1.913
times 10 to the minus 7.

1298
01:20:26,230 --> 01:20:29,960
So about 1 in 5 million.

1299
01:20:29,960 --> 01:20:32,760
The reason this gets
so small is because you

1300
01:20:32,760 --> 01:20:35,872
get the n in that
exponent up there.

1301
01:20:35,872 --> 01:20:37,330
And of course, this
is all computed

1302
01:20:37,330 --> 01:20:39,560
using Stirling's formula,
if you actually want

1303
01:20:39,560 --> 01:20:41,370
to go through the calculations.

1304
01:20:41,370 --> 01:20:46,190
You go start with n choose
k, which is your n factorial,

1305
01:20:46,190 --> 01:20:48,980
times over k factorial
n minus k factorial.

1306
01:20:48,980 --> 01:20:52,280
Plug in Stirling's formula and
you do a bunch of messy stuff,

1307
01:20:52,280 --> 01:20:54,840
and these things pop out.

1308
01:20:54,840 --> 01:20:57,720
And so it gets exponentially
small, exponentially fast.

1309
01:20:57,720 --> 01:21:00,270
In fact, if you were
to plot this thing,

1310
01:21:00,270 --> 01:21:03,510
if you plot the binomial
distribution function,

1311
01:21:03,510 --> 01:21:06,520
it looks something like this.

1312
01:21:17,330 --> 01:21:21,380
All right, so I have
0 to n, and then I

1313
01:21:21,380 --> 01:21:24,957
have pn, which is the maximum.

1314
01:21:24,957 --> 01:21:25,915
Here's the probability.

1315
01:21:25,915 --> 01:21:28,560
It goes 0 to 1.

1316
01:21:28,560 --> 01:21:33,310
And your maximum value is here
at 8% in the case of 100 coins,

1317
01:21:33,310 --> 01:21:39,980
and it just zooms down
exponential small.

1318
01:21:39,980 --> 01:21:43,110
You can't even draw because
it gets so small so fast.

1319
01:21:43,110 --> 01:21:50,840
This height here is
about 1 over root n.

1320
01:21:50,840 --> 01:21:55,920
And this width here of
the hump, is about root n.

1321
01:21:55,920 --> 01:22:01,470
And these things zoom down
to 0, exponentially fast.

1322
01:22:01,470 --> 01:22:04,220
And so that's what the binomial
distribution looks like.

1323
01:22:04,220 --> 01:22:12,110
And so it says that you are very
likely to be very close to pn

1324
01:22:12,110 --> 01:22:15,494
heads, or pn things happening.

1325
01:22:15,494 --> 01:22:17,660
And I won't go through the
math now-- probably do it

1326
01:22:17,660 --> 01:22:20,300
in recitation
tomorrow-- of computing

1327
01:22:20,300 --> 01:22:23,770
at most 25 heads, which is
the cumulative distribution

1328
01:22:23,770 --> 01:22:25,222
function.

1329
01:22:25,222 --> 01:22:26,930
And this comes up in
all sorts of places.

1330
01:22:26,930 --> 01:22:29,920
Like you have a noisy
communications channel,

1331
01:22:29,920 --> 01:22:33,620
and every bit as
dropped with 1% chance.

1332
01:22:33,620 --> 01:22:35,340
If you have 10,000
bits, you'd like

1333
01:22:35,340 --> 01:22:40,000
to know, what's the
probability I lost 2% of them

1334
01:22:40,000 --> 01:22:42,990
when I have 1% failure rate?

1335
01:22:42,990 --> 01:22:46,050
The chance of losing
2% out of 10,000

1336
01:22:46,050 --> 01:22:49,500
is like 2 to the minus 60 if
they're mutually independent.

1337
01:22:49,500 --> 01:22:52,001
So no chance of losing 2%.

1338
01:22:52,001 --> 01:22:52,876
All right, very good.

1339
01:22:52,876 --> 01:22:55,730
We'll do more this in
recitation tomorrow.