Mathematics for Computer Science

Chapter 16 Events and Probability Spaces538

set of all people

in the world

set of people

who live in

Cambridge

set of MIT

students

B

A

Figure 16.12 Selecting a random person.Ais the event that the person is an MIT
student.Bis the even that the person lives in Cambridge.

 What is the probability that two rolled dice sum to 10, given that both are

odd?

 What is the probability that I’ll get four-of-a-kind in Texas No Limit Hold

’Em Poker, given that I’m initially dealt two queens?

There is a special notation for conditional probabilities. In general, Pr



AjB



denotes the probability of eventA, given that eventBhappens. So, in our example,
Pr



AjB



is the probability that a random person is an MIT student, given that he
or she is a Cambridge resident.
How do we compute Pr



AjB



? Since we aregiventhat the person lives in
Cambridge, we can forget about everyone in the world who does not. Thus, all
outcomes outside eventBare irrelevant. So, intuitively, Pr



AjB



should be the
fraction of Cambridge residents that are also MIT students; that is, the answer
should be the probability that the person is in setA\B(the darkly shaded region
in Figure 16.12) divided by the probability that the person is in setB(the lightly
shaded region). This motivates the definition of conditional probability:

Definition 16.5.1.

Pr



AjB



WWD

PrŒA\Bç

PrŒBç

If PrŒBçD 0 , then the conditional probability Pr



AjB



is undefined.