Let us now examine what conditional
probabilities are good for.
We have already discussed that they are used to revise a
model when we get new information, but there is
another way in which they arise.
We can use conditional probabilities to build a
multi-stage model of a probabilistic experiment.
We will illustrate this through an example involving
the detection of an object up in the sky by a radar.
We will keep our example very simple.
On the other hand, it turns out to have all the basic
elements of a real-world model.
So, we are looking up in the sky, and either there's an
airplane flying up there or not.
Let us call Event A the event that an airplane is indeed
flying up there, and we have two possibilities.
Either Event A occurs, or the complement of A occurs, in
which case nothing is flying up there.
At this point, we can also assign some probabilities to
these two possibilities.
Let us say that through prior experience, perhaps, or some
other knowledge, we know that the probability that something
is indeed flying up there is 5% and with probability 95%
nothing is flying.
Now, we also have a radar that looks up there, and there are
two things that can happen.
Either something registers on the radar
screen or nothing registers.
Of course, if it's a good radar, probably Event B will
tend to go together with Event A. But it's also possible that
the radar will make some mistakes.
And so we have various possibilities.
If there's a plane up there, it's possible that the radar
will detect it, in which case Event B will also happen.
But it's also conceivable that the radar will not detect it,
in which case we have a so-called miss.
And so a plane is flying up there, but the radar missed
it, did not detect it.
Another possibility is that nothing is flying up there,
but the radar does detect something, and this is a
situation that's called a false alarm.
Finally, there's the possibility that nothing is
flying up there, and the radar did not see anything either.
Now, let us focus on this particular situation.
Suppose that Event A has occurred.
So we are living inside this particular universe.
In this universe, there are two possibilities, and we can
assign probabilities to these two possibilities.
So let's say that if something is flying up there, our radar
will find it with probability 99%, but will also miss it
with probability 1%.
What's the meaning of this number, 99%?
Well, this is a probability that applies to a situation
where an airplane is up there.
So it is really a conditional probability.
It's the conditional probability that we will
detect something, the radar will detect the plane, given
that the plane is already flying up there.
And similarly, this 1% can be thought of as the conditional
probability that the complement of B occurs, so the
radar doesn't see anything, given that there is a plane up
in the sky.
We can assign similar probabilities
under the other scenario.
If there is no plane, there is a probability that there will
be a false alarm, and there is a probability that the radar
will not see anything.
Those four numbers here are, in essence, the
specs of our radar.
They describe how the radar behaves in a world in which an
airplane has been placed in the sky, and some other
numbers that describe how the radar behaves in a world where
nothing is flying up in the sky.
So we have described various probabilistic properties of
our model, but is it a complete model?
Can we calculate anything that we might wish to calculate?
Let us look at this question.
Can we calculate the probability that
both A and B occur?
It's this particular scenario here.
How can we calculate it?
Well, let us remember the definition of conditional
probabilities.
The conditional probability of an event given another event
is the probability of their intersection divided by the
probability of the conditioning event.
But this doesn't quite help us because if we try to calculate
the numerator, we do not have the value of the probability
of A given B. We have the value of the probability of B
given A. What can we do?
Well, we notice that we can use this definition of
conditional probabilities, but use it in the reverse
direction, interchanging the roles of A and B. If we
interchange the roles of A and B, our definition leads to the
following expression.
The conditional probability of B given A is the probability
that both events occur divided by the probability, again, of
the conditioning event.
Therefore, the probability that A and B occur is equal to
the probability that A occurs times the conditional
probability that B occurs given that A occurred.
And in our example, this is 0.05 times the conditional
probability that B occurs, which is 0.99.
So we can calculate the probability of this particular
event by multiplying probabilities and conditional
probabilities along the path in this tree diagram that
leads us here.
And we can do the same for any other leaf in this diagram.
So for example, the probability that this happens
is going to be the probability of this event times the
conditional probability of B complement given that A
complement has occurred.
How about a different question?
What is the probability, the total probability, that the
radar sees something?
Let us try to identify this event.
The radar can see something under two scenarios.
There's the scenario where there is a plane up in the sky
and the radar sees it.
And there's another scenario where nothing is up in the
sky, but the radar thinks that it sees something.
So these two possibilities together make up the event B.
And so to calculate the probability of B, we need to
add the probabilities of these two events.
For the first event, we already calculated it.
It's 0.05 times 0.90.
For the second possibility, we need to do a similar
calculation.
The probability that this occurs is equal to 0.95 times
the conditional probability of B occurring under the scenario
where A complement has occurred, and this is 0.1.
If we add those two numbers together, the answer turns out
to be 0.1445.
Finally, a last question, which is perhaps the most
interesting one.
Suppose that the radar registered something.
What is the probability that there is an
airplane out there?
How do we do this calculation?
Well, we can start from the definition of the conditional
probability of A given B, and note that we already have in
our hands both the numerator and the denominator.
So the numerator is this number, 0.05 times 0.99, and
the denominator is 0.1445, and we can use our calculators to
see that the answer is approximately 0.34.
So there is a 34% probability that an airplane is there
given that the radar has seen or thinks
that it sees something.
So the numerical value of this answer is somewhat interesting
because it's pretty small.
Even though we have a very good radar that tells us the
right thing 99% of the time under one scenario and 90%
under the other scenario.
Despite that, given that the radar has seen something, this
is not really convincing or compelling evidence that there
is an airplane up there.
The probability that there's an airplane up there is only
34% in a situation where the radar thinks
that it has seen something.
So in the next few segments, we are going to revisit these
three calculations and see how they can generalize.
In fact, a large part of what is to happen in the remainder
of this class will be elaboration
on these three ideas.
They are three types of calculations that will show up
over and over, of course, in more complicated forms, but
the basic ideas are essentially captured in this
simple example.