16.5. Conditional Probability 537
where Tnstands for a lengthnstring of T’s. The probability function is
PrŒTnHçWWD
1
2 nC^1
:
To verify that this is a probability space, we just have to check that all the probabili-
ties are nonnegative and that they sum to 1. Nonnegativity is obvious, and applying
the formula for the sum of a geometric series, we find that
X
n 2 N
PrŒTnHçD
X
n 2 N
1
2 nC^1
D1:
Notice that this model does not have an outcome corresponding to the possi-
bility that both players keep flipping tails forever —in the diagram, flipping for-
ever corresponds to following the infinite path in the tree without ever reaching
a leaf/outcome. If leaving this possibility out of the model bothers you, you’re
welcome to fix it by adding another outcome,!forever, to indicate that that’s what
happened. Of course since the probabililities of the other outcomes already sum to
1, you have to define the probability of!foreverto be 0. Now outcomes with prob-
ability zero will have no impact on our calculations, so there’s no harm in adding
it in if it makes you happier. On the other hand, in countable probability spaces
it isn’t necessary to have outcomes with probability zero, and we will generally
ignore them.
16.5 Conditional Probability
Suppose that we pick a random person in the world. Everyone has an equal chance
of being selected. LetAbe the event that the person is an MIT student, and let
Bbe the event that the person lives in Cambridge. What are the probabilities of
these events? Intuitively, we’re picking a random point in the big ellipse shown in
Figure 16.12 and asking how likely that point is to fall into regionAorB.
The vast majority of people in the world neither live in Cambridge nor are MIT
students, so eventsAandBboth have low probability. But what about the prob-
ability that a person is an MIT student,giventhat the person lives in Cambridge?
This should be much greater —but what is it exactly?
What we’re asking for is called aconditional probability; that is, the probability
that one event happens, given that some other event definitely happens. Questions
about conditional probabilities come up all the time:
What is the probability that it will rain this afternoon, given that it is cloudy
this morning?
