1
00:00:00,170 --> 00:00:03,400
We will now go through a
probability calculation for

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00:00:03,400 --> 00:00:06,750
the case where we have a
continuous sample space.

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00:00:06,750 --> 00:00:10,820
We revisit our earlier example
in which we were throwing a

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00:00:10,820 --> 00:00:14,840
dart into a square target,
the square target

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00:00:14,840 --> 00:00:16,010
being the unit square.

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00:00:16,010 --> 00:00:18,970
And we were guaranteed that our
dart would fall somewhere

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00:00:18,970 --> 00:00:20,780
inside this set.

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00:00:20,780 --> 00:00:24,090
So our sample space is the
unit square itself.

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00:00:24,090 --> 00:00:27,670
We have a description of the
sample space, but we do not

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00:00:27,670 --> 00:00:29,140
yet have a probability law.

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00:00:29,140 --> 00:00:31,340
We need to specify one.

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00:00:31,340 --> 00:00:35,700
The choice of a probability
law could be arbitrary.

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00:00:35,700 --> 00:00:39,620
It's up to us to choose how to
model a certain situation.

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00:00:39,620 --> 00:00:42,780
And to keep things simple, we're
going to assume that our

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00:00:42,780 --> 00:00:48,030
probability law is a uniform
one, which means that the

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00:00:48,030 --> 00:00:51,350
probability of any particular
subset of the sample space is

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00:00:51,350 --> 00:00:54,040
going to be the area
of that subset.

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00:00:54,040 --> 00:00:57,770
So if we have some subset lying
somewhere here and we

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00:00:57,770 --> 00:01:01,190
ask what is the probability that
we fall into that subset?

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00:01:01,190 --> 00:01:04,470
The probability is exactly
the area of

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00:01:04,470 --> 00:01:07,010
that particular subset.

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00:01:07,010 --> 00:01:09,860
Once more, this is an arbitrary
choice of a

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00:01:09,860 --> 00:01:11,000
probability law.

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00:01:11,000 --> 00:01:14,140
There's nothing in our
assumptions so far that would

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00:01:14,140 --> 00:01:16,370
force us to make this
particular choice.

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00:01:16,370 --> 00:01:20,300
And we just use it for the
purposes of this example.

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00:01:20,300 --> 00:01:24,370
So now let us calculate
some probabilities.

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00:01:24,370 --> 00:01:26,750
Let us look at this event.

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00:01:26,750 --> 00:01:32,539
This is the event that the sum
of the two numbers that we get

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00:01:32,539 --> 00:01:36,440
in our experiment is less
than or equal to 1/2.

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00:01:36,440 --> 00:01:40,090
It is always useful to work in
terms of a picture and to

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00:01:40,090 --> 00:01:44,259
depict that event in a picture
of the sample space.

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00:01:44,259 --> 00:01:48,640
So in terms of that sample
space, the points that make

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00:01:48,640 --> 00:01:55,900
this event to be true are just a
triangle that lies below the

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00:01:55,900 --> 00:02:02,940
line, where this is the line,
that's x plus y equals 1/2.

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00:02:02,940 --> 00:02:06,860
Anything below that line, these
are the outcomes that

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00:02:06,860 --> 00:02:08,978
make this event happen.

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00:02:08,978 --> 00:02:12,360
So we're trying to find the
probability of this red event.

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00:02:12,360 --> 00:02:16,040
We have assumed that probability
is equal to area.

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00:02:16,040 --> 00:02:19,310
Therefore, the probability we're
trying to calculate is

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00:02:19,310 --> 00:02:21,180
the area of a triangle.

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00:02:21,180 --> 00:02:25,880
And the area of a triangle is
1/2 times the base of the

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00:02:25,880 --> 00:02:30,220
triangle, which is 1/2 in our
case, times the height of the

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00:02:30,220 --> 00:02:33,650
triangle, which is again
1/2 in our case.

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00:02:33,650 --> 00:02:35,820
And the end result is 1/8.

46
00:02:38,850 --> 00:02:43,390
Let us now calculate another
probability.

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00:02:43,390 --> 00:02:46,730
Now, this is an event
that consists of

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00:02:46,730 --> 00:02:49,240
only a single element.

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00:02:49,240 --> 00:02:55,760
We take the point 0.5, 0.3,
which sits somewhere here.

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00:02:55,760 --> 00:03:00,220
The event of interest is a set,
but that set consists of

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00:03:00,220 --> 00:03:01,380
a single point.

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00:03:01,380 --> 00:03:04,260
So we're asking for the
probability that our dart

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00:03:04,260 --> 00:03:08,470
falls exactly on top
of that point.

54
00:03:08,470 --> 00:03:09,210
What is it?

55
00:03:09,210 --> 00:03:12,260
Well, it is the area
of a set that

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00:03:12,260 --> 00:03:13,750
consists of a single point.

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00:03:13,750 --> 00:03:16,050
What is the area of
a single point?

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00:03:16,050 --> 00:03:17,910
It is 0.

59
00:03:17,910 --> 00:03:20,990
And similarly for any other
single point inside that

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00:03:20,990 --> 00:03:24,090
sample space that we might have
considered, the answer is

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00:03:24,090 --> 00:03:27,500
going to be 0.

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00:03:27,500 --> 00:03:31,430
Let us now abstract from this
example, as well as the

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00:03:31,430 --> 00:03:34,850
previous one, and note
the following.

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00:03:34,850 --> 00:03:37,210
Probability calculations
involve a

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00:03:37,210 --> 00:03:40,050
sequence of four steps.

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00:03:40,050 --> 00:03:44,000
Starting with a word description
of a problem, of a

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00:03:44,000 --> 00:03:46,670
probabilistic experiment,
we first write

68
00:03:46,670 --> 00:03:49,200
down the sample space.

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00:03:49,200 --> 00:03:52,490
Then we specify a
probability law.

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00:03:52,490 --> 00:03:55,270
Let me emphasize again here
that this step has some

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00:03:55,270 --> 00:03:57,030
arbitrariness in it.

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00:03:57,030 --> 00:03:59,840
You can choose any probability
law you like, although for

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00:03:59,840 --> 00:04:02,480
your results to be useful
it would be good if your

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00:04:02,480 --> 00:04:05,350
probability law captures the
real-world phenomenon you're

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00:04:05,350 --> 00:04:07,090
trying to model.

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00:04:07,090 --> 00:04:11,010
Typically you're interested in
calculating the probability of

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00:04:11,010 --> 00:04:13,170
some event.

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00:04:13,170 --> 00:04:16,519
That event may be described in
some loose manner, so you need

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00:04:16,519 --> 00:04:18,720
to describe it mathematically.

80
00:04:18,720 --> 00:04:22,570
And if possible, it's always
good to describe it in terms

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00:04:22,570 --> 00:04:23,690
of a picture.

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00:04:23,690 --> 00:04:26,300
Pictures are immensely
useful when going

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00:04:26,300 --> 00:04:28,350
through this process.

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00:04:28,350 --> 00:04:32,970
And finally, the last step is to
go ahead and calculate the

85
00:04:32,970 --> 00:04:35,890
probability of the event
of interest.

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00:04:35,890 --> 00:04:38,690
Now, a probability law in
principle specifies the

87
00:04:38,690 --> 00:04:40,930
probability of every
event, and there's

88
00:04:40,930 --> 00:04:42,860
nothing else to do.

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00:04:42,860 --> 00:04:45,670
But quite often the probability
law will be given

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00:04:45,670 --> 00:04:49,540
in some implicit manner, for
example, by specifying the

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00:04:49,540 --> 00:04:52,390
probabilities of only
some of the events.

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00:04:52,390 --> 00:04:55,310
In that case, you may have to
do some additional work to

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00:04:55,310 --> 00:04:58,010
find the probability of
the particular event

94
00:04:58,010 --> 00:04:59,870
that you care about.

95
00:04:59,870 --> 00:05:03,840
This last step sometimes
will be easy.

96
00:05:03,840 --> 00:05:06,340
Sometimes it may
be complicated.

97
00:05:06,340 --> 00:05:10,820
But in either case, by following
this four-step

98
00:05:10,820 --> 00:05:14,790
procedure and by being
systematic you will always be

99
00:05:14,790 --> 00:05:17,530
able to come up with a single
correct answer.