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Let us now examine
what conditional

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probabilities are good for.

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We have already discussed that
they are used to revise a

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model when we get new
information, but there is

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another way in which
they arise.

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We can use conditional
probabilities to build a

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multi-stage model of a
probabilistic experiment.

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We will illustrate this through
an example involving

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the detection of an object
up in the sky by a radar.

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We will keep our example
very simple.

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On the other hand, it turns
out to have all the basic

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elements of a real-world
model.

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So, we are looking up in the
sky, and either there's an

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airplane flying up
there or not.

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Let us call Event A the event
that an airplane is indeed

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flying up there, and we have
two possibilities.

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Either Event A occurs, or the
complement of A occurs, in

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which case nothing is
flying up there.

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At this point, we can also
assign some probabilities to

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these two possibilities.

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Let us say that through prior
experience, perhaps, or some

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other knowledge, we know that
the probability that something

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is indeed flying up there is
5% and with probability 95%

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nothing is flying.

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Now, we also have a radar that
looks up there, and there are

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two things that can happen.

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Either something registers
on the radar

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screen or nothing registers.

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Of course, if it's a good radar,
probably Event B will

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tend to go together with Event
A. But it's also possible that

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the radar will make
some mistakes.

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And so we have various
possibilities.

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If there's a plane up there,
it's possible that the radar

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will detect it, in which case
Event B will also happen.

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00:02:01,350 --> 00:02:05,110
But it's also conceivable that
the radar will not detect it,

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in which case we have
a so-called miss.

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And so a plane is flying up
there, but the radar missed

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it, did not detect it.

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Another possibility is that
nothing is flying up there,

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but the radar does detect
something, and this is a

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situation that's called
a false alarm.

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Finally, there's the possibility
that nothing is

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flying up there, and the radar
did not see anything either.

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Now, let us focus on this
particular situation.

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Suppose that Event
A has occurred.

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So we are living inside this
particular universe.

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In this universe, there are two
possibilities, and we can

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assign probabilities to these
two possibilities.

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00:02:58,360 --> 00:03:01,920
So let's say that if something
is flying up there, our radar

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will find it with probability
99%, but will also miss it

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with probability 1%.

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What's the meaning of
this number, 99%?

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Well, this is a probability that
applies to a situation

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where an airplane is up there.

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So it is really a conditional
probability.

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It's the conditional probability
that we will

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detect something, the radar will
detect the plane, given

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that the plane is already
flying up there.

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And similarly, this 1% can be
thought of as the conditional

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probability that the complement
of B occurs, so the

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radar doesn't see anything,
given that there is a plane up

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in the sky.

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We can assign similar
probabilities

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under the other scenario.

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If there is no plane, there is
a probability that there will

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be a false alarm, and there is
a probability that the radar

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will not see anything.

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Those four numbers here
are, in essence, the

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specs of our radar.

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They describe how the radar
behaves in a world in which an

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airplane has been placed in
the sky, and some other

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numbers that describe how the
radar behaves in a world where

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nothing is flying
up in the sky.

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So we have described various
probabilistic properties of

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our model, but is it
a complete model?

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Can we calculate anything that
we might wish to calculate?

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Let us look at this question.

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Can we calculate the
probability that

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both A and B occur?

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It's this particular
scenario here.

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How can we calculate it?

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Well, let us remember the
definition of conditional

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probabilities.

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The conditional probability of
an event given another event

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is the probability of their
intersection divided by the

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00:05:02,340 --> 00:05:05,860
probability of the conditioning
event.

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00:05:05,860 --> 00:05:12,480
But this doesn't quite help us
because if we try to calculate

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the numerator, we do not have
the value of the probability

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00:05:16,180 --> 00:05:19,610
of A given B. We have the value
of the probability of B

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00:05:19,610 --> 00:05:22,540
given A. What can we do?

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Well, we notice that we can
use this definition of

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conditional probabilities,
but use it in the reverse

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direction, interchanging the
roles of A and B. If we

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interchange the roles of A and
B, our definition leads to the

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following expression.

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The conditional probability of
B given A is the probability

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that both events occur divided
by the probability, again, of

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the conditioning event.

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00:05:50,330 --> 00:05:56,070
Therefore, the probability that
A and B occur is equal to

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the probability that A occurs
times the conditional

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00:06:00,880 --> 00:06:06,240
probability that B occurs
given that A occurred.

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And in our example, this is
0.05 times the conditional

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probability that B occurs,
which is 0.99.

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So we can calculate the
probability of this particular

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00:06:23,400 --> 00:06:28,930
event by multiplying
probabilities and conditional

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probabilities along the path
in this tree diagram that

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leads us here.

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And we can do the same for any
other leaf in this diagram.

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So for example, the probability
that this happens

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is going to be the probability
of this event times the

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conditional probability of
B complement given that A

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complement has occurred.

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How about a different
question?

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What is the probability, the
total probability, that the

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00:07:00,570 --> 00:07:03,450
radar sees something?

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Let us try to identify
this event.

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The radar can see something
under two scenarios.

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There's the scenario where there
is a plane up in the sky

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and the radar sees it.

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And there's another scenario
where nothing is up in the

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sky, but the radar thinks
that it sees something.

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So these two possibilities
together make up the event B.

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And so to calculate the
probability of B, we need to

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add the probabilities
of these two events.

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For the first event, we
already calculated it.

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It's 0.05 times 0.90.

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For the second possibility,
we need to do a similar

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calculation.

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00:07:52,480 --> 00:07:58,820
The probability that this occurs
is equal to 0.95 times

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the conditional probability of B
occurring under the scenario

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where A complement has occurred,
and this is 0.1.

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If we add those two numbers
together, the answer turns out

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to be 0.1445.

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Finally, a last question, which
is perhaps the most

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interesting one.

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Suppose that the radar
registered something.

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What is the probability
that there is an

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airplane out there?

139
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How do we do this calculation?

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Well, we can start from the
definition of the conditional

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probability of A given B, and
note that we already have in

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our hands both the numerator
and the denominator.

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00:08:58,880 --> 00:09:08,780
So the numerator is this number,
0.05 times 0.99, and

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00:09:08,780 --> 00:09:15,920
the denominator is 0.1445, and
we can use our calculators to

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see that the answer is
approximately 0.34.

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00:09:21,800 --> 00:09:27,650
So there is a 34% probability
that an airplane is there

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00:09:27,650 --> 00:09:33,330
given that the radar
has seen or thinks

148
00:09:33,330 --> 00:09:36,530
that it sees something.

149
00:09:36,530 --> 00:09:42,220
So the numerical value of this
answer is somewhat interesting

150
00:09:42,220 --> 00:09:44,100
because it's pretty small.

151
00:09:44,100 --> 00:09:48,100
Even though we have a very good
radar that tells us the

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00:09:48,100 --> 00:09:54,320
right thing 99% of the time
under one scenario and 90%

153
00:09:54,320 --> 00:09:56,630
under the other scenario.

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00:09:56,630 --> 00:10:01,370
Despite that, given that the
radar has seen something, this

155
00:10:01,370 --> 00:10:07,000
is not really convincing or
compelling evidence that there

156
00:10:07,000 --> 00:10:08,890
is an airplane up there.

157
00:10:08,890 --> 00:10:12,600
The probability that there's an
airplane up there is only

158
00:10:12,600 --> 00:10:16,590
34% in a situation where
the radar thinks

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00:10:16,590 --> 00:10:20,830
that it has seen something.

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00:10:20,830 --> 00:10:26,310
So in the next few segments, we
are going to revisit these

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00:10:26,310 --> 00:10:31,601
three calculations and see
how they can generalize.

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00:10:31,601 --> 00:10:37,640
In fact, a large part of what is
to happen in the remainder

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00:10:37,640 --> 00:10:40,430
of this class will
be elaboration

164
00:10:40,430 --> 00:10:42,770
on these three ideas.

165
00:10:42,770 --> 00:10:46,960
They are three types of
calculations that will show up

166
00:10:46,960 --> 00:10:51,460
over and over, of course, in
more complicated forms, but

167
00:10:51,460 --> 00:10:54,970
the basic ideas are essentially
captured in this

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00:10:54,970 --> 00:10:56,220
simple example.