WEBVTT
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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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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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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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probability of the conditioning
event.
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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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of A given B. We have the value
of the probability of B
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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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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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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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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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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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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?
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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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So the numerator is this number,
0.05 times 0.99, and
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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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So there is a 34% probability
that an airplane is there
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given that the radar
has seen or thinks
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that it sees something.
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So the numerical value of this
answer is somewhat interesting
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because it's pretty small.
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Even though we have a very good
radar that tells us the
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right thing 99% of the time
under one scenario and 90%
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under the other scenario.
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Despite that, given that the
radar has seen something, this
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is not really convincing or
compelling evidence that there
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is an airplane up there.
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The probability that there's an
airplane up there is only
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34% in a situation where
the radar thinks
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that it has seen something.
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So in the next few segments, we
are going to revisit these
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three calculations and see
how they can generalize.
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In fact, a large part of what is
to happen in the remainder
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of this class will
be elaboration
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on these three ideas.
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They are three types of
calculations that will show up
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over and over, of course, in
more complicated forms, but
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the basic ideas are essentially
captured in this
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simple example.