1
00:00:00,409 --> 00:00:02,450
PROFESSOR: One of the most
important applications

2
00:00:02,450 --> 00:00:05,720
of conditional probability
is in analyzing

3
00:00:05,720 --> 00:00:10,260
the results of diagnostic
tests of uncertain reliability.

4
00:00:10,260 --> 00:00:14,520
So let's look at a
fundamental example.

5
00:00:14,520 --> 00:00:19,082
Suppose that I have a diagnostic
test for tuberculosis.

6
00:00:19,082 --> 00:00:20,540
It really sounds
great because it's

7
00:00:20,540 --> 00:00:23,710
going to be 99% accurate--
in fact, more than 99%

8
00:00:23,710 --> 00:00:25,250
accurate, really,
because here are

9
00:00:25,250 --> 00:00:27,050
the properties this test has.

10
00:00:27,050 --> 00:00:30,710
If you have TB, this test
is guaranteed to detect it

11
00:00:30,710 --> 00:00:33,190
and say, yes, you have TB.

12
00:00:33,190 --> 00:00:36,660
If you don't have
TB, 99% of the time,

13
00:00:36,660 --> 00:00:40,550
the test says correctly
that you don't

14
00:00:40,550 --> 00:00:45,390
have TB, and 1% of the
time, it gets it wrong.

15
00:00:45,390 --> 00:00:48,520
Now, suppose the doctor gives
you the test and the test

16
00:00:48,520 --> 00:00:50,630
comes up saying
that you have TB.

17
00:00:50,630 --> 00:00:54,110
That's kind of scary
because TB is, in fact,

18
00:00:54,110 --> 00:00:55,180
quite a serious disease.

19
00:00:55,180 --> 00:00:56,680
It's getting worse
because there are

20
00:00:56,680 --> 00:00:59,250
all of these
antibiotic-resistant versions

21
00:00:59,250 --> 00:01:00,030
of TB.

22
00:01:00,030 --> 00:01:04,110
Now in Asia, where all the
known antibiotics are not

23
00:01:04,110 --> 00:01:08,410
very effective-- if effective
at all-- of curing it,

24
00:01:08,410 --> 00:01:12,370
and this test that
was 99% accurate

25
00:01:12,370 --> 00:01:16,090
says I have this disease,
it sounds really worrisome.

26
00:01:16,090 --> 00:01:19,652
But in fact, we can
ask more technically,

27
00:01:19,652 --> 00:01:20,860
should you really be worried?

28
00:01:20,860 --> 00:01:24,930
What is the probability given
that this apparently highly

29
00:01:24,930 --> 00:01:26,700
accurate test says you have TB?

30
00:01:26,700 --> 00:01:28,800
What's the probability
that you actually have TB?

31
00:01:28,800 --> 00:01:30,630
That's what we
want to calculate.

32
00:01:30,630 --> 00:01:33,220
What's the probability
that you have it?

33
00:01:33,220 --> 00:01:36,900
So in other words, I want
the conditional probability

34
00:01:36,900 --> 00:01:39,825
that I have TB given that
the test comes in positive.

35
00:01:39,825 --> 00:01:44,210
The test says, yes, you have TB.

36
00:01:44,210 --> 00:01:47,114
That test positive is a big
word that I won't have room

37
00:01:47,114 --> 00:01:48,530
for on other slides,
so let's just

38
00:01:48,530 --> 00:01:50,380
abbreviate it by
[? plus. ?] Plus means,

39
00:01:50,380 --> 00:01:56,470
in green, that the test said,
yes, positive-- you have TB.

40
00:01:56,470 --> 00:01:58,790
OK, so that's the
probability that we're

41
00:01:58,790 --> 00:02:01,090
trying to calculate, this
conditional probability.

42
00:02:01,090 --> 00:02:02,440
What do we know about the test?

43
00:02:02,440 --> 00:02:04,030
Let's translate
the information we

44
00:02:04,030 --> 00:02:05,950
have about the test
into the language

45
00:02:05,950 --> 00:02:07,400
of conditional probability.

46
00:02:07,400 --> 00:02:10,210
And the first thing we said
was that the test is guaranteed

47
00:02:10,210 --> 00:02:11,980
to get it right if you have TB.

48
00:02:11,980 --> 00:02:15,560
So given that you have
TB, the probability

49
00:02:15,560 --> 00:02:18,380
that the test will say so-- it
will return a positive result--

50
00:02:18,380 --> 00:02:21,060
is 1.

51
00:02:21,060 --> 00:02:23,820
Given that you don't
have TB, the probability

52
00:02:23,820 --> 00:02:27,370
that the test will say that you
do have TB is only 1 in 100.

53
00:02:27,370 --> 00:02:29,920
Because 99% of the
time, it correctly

54
00:02:29,920 --> 00:02:32,030
says you don't have TB.

55
00:02:32,030 --> 00:02:35,530
And 1% of the time, it
says oops, you do have TB.

56
00:02:35,530 --> 00:02:37,800
So this is what's called
a false positive rate.

57
00:02:37,800 --> 00:02:41,510
It's falsely claiming that you
have TB when you really don't.

58
00:02:41,510 --> 00:02:46,430
And that rate, we're
hypothesizing, is only 1%.

59
00:02:46,430 --> 00:02:48,150
Now, what we're
trying to calculate,

60
00:02:48,150 --> 00:02:51,870
again, is the probability
that you have TB

61
00:02:51,870 --> 00:02:56,494
given that the test came in
positive and said you had TB.

62
00:02:56,494 --> 00:02:57,910
Well, let's look
at the definition

63
00:02:57,910 --> 00:02:59,034
of conditional probability.

64
00:02:59,034 --> 00:03:02,260
The probability that you have
TB given that the test came

65
00:03:02,260 --> 00:03:07,050
in positive, that said you
do, is simply the probability

66
00:03:07,050 --> 00:03:09,220
that both the test
comes in positive

67
00:03:09,220 --> 00:03:12,640
and you have TB divided
by the probability

68
00:03:12,640 --> 00:03:15,330
that the test comes in positive.

69
00:03:15,330 --> 00:03:20,110
Well, using the definition of
conditional probability again,

70
00:03:20,110 --> 00:03:25,580
this intersection, this AND
of having TB and the test

71
00:03:25,580 --> 00:03:28,040
coming in positive, is
simply the probability

72
00:03:28,040 --> 00:03:30,070
that the test comes
in positive given

73
00:03:30,070 --> 00:03:34,680
that you have TB times the
probability that you have TB.

74
00:03:34,680 --> 00:03:36,330
Now, this one we know.

75
00:03:36,330 --> 00:03:38,390
It's 1 because the
test is perfect.

76
00:03:38,390 --> 00:03:39,980
If you have TB, the
test is definitely

77
00:03:39,980 --> 00:03:41,200
going to say positive.

78
00:03:41,200 --> 00:03:43,600
So that lets me
simplify things nicely.

79
00:03:43,600 --> 00:03:46,220
What I've just figured
out is the probability

80
00:03:46,220 --> 00:03:49,860
that you have TB given
that the test says

81
00:03:49,860 --> 00:03:53,180
you do is simply the
quotient of the probability

82
00:03:53,180 --> 00:03:57,570
that you have TB given no other
information and the probability

83
00:03:57,570 --> 00:04:01,310
that the test comes in positive.

84
00:04:01,310 --> 00:04:03,490
Well, what is that
probability that the test

85
00:04:03,490 --> 00:04:04,240
comes in positive?

86
00:04:04,240 --> 00:04:05,698
How are we going
to calculate that?

87
00:04:05,698 --> 00:04:08,197
That's the key unknown here.

88
00:04:08,197 --> 00:04:10,030
And we're going to use
the probability rule,

89
00:04:10,030 --> 00:04:11,730
the total probability rule.

90
00:04:11,730 --> 00:04:14,700
Total probability says that
you do or you don't have TB.

91
00:04:14,700 --> 00:04:20,240
So that the way to calculate the
probability that the test comes

92
00:04:20,240 --> 00:04:22,700
in positive is to look
at the probability

93
00:04:22,700 --> 00:04:26,110
that the test comes in positive
when you do and don't have TB.

94
00:04:26,110 --> 00:04:28,040
And we know those numbers.

95
00:04:28,040 --> 00:04:31,600
So let's look at the
total probability formula.

96
00:04:31,600 --> 00:04:33,900
The probability that the
test comes in positive

97
00:04:33,900 --> 00:04:36,710
is simply the probability
that it comes in positive

98
00:04:36,710 --> 00:04:40,020
if you have TB times the
probability you have TB,

99
00:04:40,020 --> 00:04:42,470
plus the probability
it comes in positive

100
00:04:42,470 --> 00:04:45,540
given that you don't have TB
times the probability you don't

101
00:04:45,540 --> 00:04:46,690
have TB.

102
00:04:46,690 --> 00:04:48,570
Well, we know a
lot of these terms.

103
00:04:48,570 --> 00:04:50,005
Let's work them out.

104
00:04:50,005 --> 00:04:53,350
Well, the probability the
test comes in positive

105
00:04:53,350 --> 00:04:56,110
given that you have TB is 1.

106
00:04:56,110 --> 00:04:57,830
And the probability
that the test

107
00:04:57,830 --> 00:05:01,059
comes in positive given that
you don't have TB is 1/100.

108
00:05:01,059 --> 00:05:02,350
That's the false positive rate.

109
00:05:02,350 --> 00:05:03,720
We figured that already.

110
00:05:03,720 --> 00:05:06,120
What about the probability
that you don't have TB?

111
00:05:06,120 --> 00:05:08,560
Well, that's simply 1
minus the probability

112
00:05:08,560 --> 00:05:10,560
that you do have TB.

113
00:05:10,560 --> 00:05:12,580
Now I have this nice
arithmetic formula

114
00:05:12,580 --> 00:05:14,490
in the probability of TB.

115
00:05:14,490 --> 00:05:20,250
So I wind up with
the probability

116
00:05:20,250 --> 00:05:25,450
of TB plus 1/100
minus [? 1/100 ?]

117
00:05:25,450 --> 00:05:26,750
of the probability of TB.

118
00:05:26,750 --> 00:05:30,270
It leaves me with 1/100
plus the 99/100 of TB.

119
00:05:30,270 --> 00:05:32,360
So that's what
this simplifies to.

120
00:05:32,360 --> 00:05:34,040
The probability
that the test comes

121
00:05:34,040 --> 00:05:37,100
in positive given
no other information

122
00:05:37,100 --> 00:05:41,410
is 99/100 of the
probability that a person

123
00:05:41,410 --> 00:05:44,960
has TB plus 1/100.

124
00:05:44,960 --> 00:05:47,740
We'll come back to this formula.

125
00:05:47,740 --> 00:05:50,820
Well, we were working
on the probability

126
00:05:50,820 --> 00:05:52,932
that you have TB given
the test came in positive.

127
00:05:52,932 --> 00:05:54,640
We figured out that
it was this quotient.

128
00:05:54,640 --> 00:05:56,910
And now, I know what
the denominator is.

129
00:05:56,910 --> 00:06:00,850
The denominator is 99/100
times the probability

130
00:06:00,850 --> 00:06:02,070
of TB plus 1/100.

131
00:06:02,070 --> 00:06:04,780
Multiply numerator and
denominator through by 100,

132
00:06:04,780 --> 00:06:07,970
and you get that the
probability that you have TB

133
00:06:07,970 --> 00:06:11,730
given that the test says you
do is 100 times the probability

134
00:06:11,730 --> 00:06:16,450
that you have TB divided by 99
times the probability that you

135
00:06:16,450 --> 00:06:18,580
have TB plus 1.

136
00:06:18,580 --> 00:06:20,870
So let's hold formula.

137
00:06:20,870 --> 00:06:24,320
Notice the key unknown
here is the probability

138
00:06:24,320 --> 00:06:28,720
that you have TB independent
of the test, the probability

139
00:06:28,720 --> 00:06:31,680
that a random person in
the population has TB.

140
00:06:31,680 --> 00:06:34,400
If we can figure that out
or if we can look that up,

141
00:06:34,400 --> 00:06:36,470
then we know what
this unknown is,

142
00:06:36,470 --> 00:06:39,550
the probability have TB given
that the test says you do.

143
00:06:39,550 --> 00:06:45,000
Well, what is the probability
that a random person has TB?

144
00:06:45,000 --> 00:06:49,840
Well, there were 11,000 cases of
TB reported in 2011, according

145
00:06:49,840 --> 00:06:53,510
to the Center for Disease
Control in the United States.

146
00:06:53,510 --> 00:06:55,170
And you can assume
that there's going

147
00:06:55,170 --> 00:06:58,200
to be a lot of unreported
cases if there are 11,000

148
00:06:58,200 --> 00:06:59,940
reported ones, because
a lot of people

149
00:06:59,940 --> 00:07:02,820
don't even know they
have the disease.

150
00:07:02,820 --> 00:07:05,930
So let's estimate, on that
basis given that the population

151
00:07:05,930 --> 00:07:13,310
of the US is around 350 million,
that the probability of TB is

152
00:07:13,310 --> 00:07:16,000
about 1/10,000.

153
00:07:16,000 --> 00:07:18,490
Let's plug that
into our formula.

154
00:07:18,490 --> 00:07:21,510
The probability that you have
TB given the test is positive

155
00:07:21,510 --> 00:07:22,500
is this formula.

156
00:07:22,500 --> 00:07:25,510
When I plug in 1/10,000 for TB.

157
00:07:25,510 --> 00:07:31,760
I get 100/10,000 and
99/10,000 plus 1.

158
00:07:31,760 --> 00:07:35,200
Well now, I can see that the
denominator is essentially 1.

159
00:07:35,200 --> 00:07:36,800
It's 1.01.

160
00:07:36,800 --> 00:07:38,370
And the numerator is 1/100.

161
00:07:38,370 --> 00:07:41,530
And this is basically
about 1/100.

162
00:07:41,530 --> 00:07:46,370
In other words, it's not
very likely that you have TB.

163
00:07:46,370 --> 00:07:49,720
Because of the relatively
high false positive rate

164
00:07:49,720 --> 00:07:53,930
that was relatively high of
1%, that false positive rate

165
00:07:53,930 --> 00:08:01,180
washed out the actual number of
TB cases, which the TB rate was

166
00:08:01,180 --> 00:08:07,590
only 0.01%, so that almost
all of the reports of TB

167
00:08:07,590 --> 00:08:10,390
were caused by the high
false positive rate.

168
00:08:10,390 --> 00:08:14,350
And that means that when you
have a report that you've

169
00:08:14,350 --> 00:08:18,180
got TB, you still only have
a 1% chance that you actually

170
00:08:18,180 --> 00:08:20,590
have the TB.

171
00:08:20,590 --> 00:08:23,860
So the 99% accurate
test was not very useful

172
00:08:23,860 --> 00:08:26,725
here for you to figure out
what kind of action to take

173
00:08:26,725 --> 00:08:28,100
and what kind of
medicine to take

174
00:08:28,100 --> 00:08:31,350
or treatment to take given
that the test came in positive.

175
00:08:31,350 --> 00:08:33,310
With 1 in 100
chance, the odds are

176
00:08:33,310 --> 00:08:34,809
you won't do anything,
in which case

177
00:08:34,809 --> 00:08:37,850
you can wonder why your
doctor gave you the test.

178
00:08:37,850 --> 00:08:40,330
Well, the 99% test sounds good.

179
00:08:40,330 --> 00:08:41,620
We figured out that it isn't.

180
00:08:41,620 --> 00:08:46,330
And a hint about why 99%
accurate isn't really so useful

181
00:08:46,330 --> 00:08:50,290
is that there's an obvious
test that's 99.99% accurate.

182
00:08:50,290 --> 00:08:51,600
What's the test?

183
00:08:51,600 --> 00:08:52,990
Always say no.

184
00:08:52,990 --> 00:08:56,390
After all, the probability
is only 1 in 10,000

185
00:08:56,390 --> 00:08:57,890
that you're going to be wrong.

186
00:08:57,890 --> 00:09:02,150
And that's the 99.99% rate.

187
00:09:02,150 --> 00:09:05,530
So it sounds as though this
test is really worthless.

188
00:09:05,530 --> 00:09:06,600
But no, it's not.

189
00:09:06,600 --> 00:09:10,754
If you think about it a
little bit, it will be useful.

190
00:09:10,754 --> 00:09:12,170
And I'll explain
that in a minute.

191
00:09:12,170 --> 00:09:13,711
I forgot I'm getting
ahead of myself.

192
00:09:13,711 --> 00:09:17,050
Because the basic formula
that we used here was we

193
00:09:17,050 --> 00:09:20,380
figured out what the
probability of TB

194
00:09:20,380 --> 00:09:24,030
given that the test
said you had TB in terms

195
00:09:24,030 --> 00:09:26,490
of the inverse probabilities
which we knew-- that is,

196
00:09:26,490 --> 00:09:29,560
the probability that the
test came in positive given

197
00:09:29,560 --> 00:09:31,710
that you had TB.

198
00:09:31,710 --> 00:09:34,085
This is an example of what's
a famous rule in probability

199
00:09:34,085 --> 00:09:34,585
theory.

200
00:09:34,585 --> 00:09:36,740
It's called Bayes'
rule, or Bayes' law.

201
00:09:36,740 --> 00:09:37,590
And this is it.

202
00:09:37,590 --> 00:09:40,270
It's just stated in terms
of arbitrary events A and B.

203
00:09:40,270 --> 00:09:42,200
It expresses the
probability of B

204
00:09:42,200 --> 00:09:44,920
given A in terms of the
probability of A given

205
00:09:44,920 --> 00:09:48,967
B and the probabilities
of A and B independently.

206
00:09:48,967 --> 00:09:50,800
Now, I can actually
never remember this law,

207
00:09:50,800 --> 00:09:52,350
but I re-derive it
right every time

208
00:09:52,350 --> 00:09:55,910
I need to do it as we've
done in the previous slides.

209
00:09:55,910 --> 00:09:57,760
It's really a quite
straightforward law

210
00:09:57,760 --> 00:09:59,890
to derive and prove.

211
00:09:59,890 --> 00:10:02,784
But let's go back to this
99% accurate test that

212
00:10:02,784 --> 00:10:04,950
seemed worthless since there
was a trivial test that

213
00:10:04,950 --> 00:10:07,820
was 99.9% accurate.

214
00:10:07,820 --> 00:10:10,400
But in fact, it's really
helpful because it

215
00:10:10,400 --> 00:10:12,590
did increase the
probability that you

216
00:10:12,590 --> 00:10:14,770
had TB by a factor of 100.

217
00:10:14,770 --> 00:10:17,429
Before you took the test and
before you know anything,

218
00:10:17,429 --> 00:10:18,970
you thought that
your probability was

219
00:10:18,970 --> 00:10:21,806
the same as everybody
elses-- about 1 in 10,000.

220
00:10:21,806 --> 00:10:23,180
Now the test says
the probability

221
00:10:23,180 --> 00:10:25,566
that you have TB is 1 in 100.

222
00:10:25,566 --> 00:10:27,040
That's a hundred times larger.

223
00:10:27,040 --> 00:10:28,560
What's the value of that?

224
00:10:28,560 --> 00:10:32,990
Well, suppose you only had
5 million doses of medicine

225
00:10:32,990 --> 00:10:36,310
to treat this American
population of 350

226
00:10:36,310 --> 00:10:37,990
million people.

227
00:10:37,990 --> 00:10:39,950
Who should you medicate?

228
00:10:39,950 --> 00:10:43,910
Well, if you medicated a
random 5 million people

229
00:10:43,910 --> 00:10:46,230
out of 350 million,
the likelihood

230
00:10:46,230 --> 00:10:49,840
that you're going to get very
many of the real TB cases

231
00:10:49,840 --> 00:10:50,860
is small.

232
00:10:50,860 --> 00:10:53,440
It's only going to
be about 1 in 30.

233
00:10:53,440 --> 00:10:56,050
You'll only get about
1/30 of the cases.

234
00:10:56,050 --> 00:11:00,710
But if you use your
5 million doses

235
00:11:00,710 --> 00:11:05,620
to medicate the 3.5 million
people who would test positive

236
00:11:05,620 --> 00:11:10,660
under this 99%
accurate test, then

237
00:11:10,660 --> 00:11:12,760
when you test all
350 million people,

238
00:11:12,760 --> 00:11:15,560
you're going to get about 3.5
million who test positive.

239
00:11:15,560 --> 00:11:17,940
You have enough medication
to treat all of them.

240
00:11:17,940 --> 00:11:19,470
And if you treat
all of them, you're

241
00:11:19,470 --> 00:11:23,720
almost certain to get all of
the actual TB cases, all 10,000

242
00:11:23,720 --> 00:11:24,730
of them.

243
00:11:24,730 --> 00:11:28,350
So the 99% accurate test
does have an important use

244
00:11:28,350 --> 00:11:30,240
in this final
setting, a lot more

245
00:11:30,240 --> 00:11:35,640
so then the 99.99% accurate
test that simply always said

246
00:11:35,640 --> 00:11:38,530
no-- food for thought.