Discrete Mathematics for Computer Science

Independent Events and Conditional Probability 511

simply divide the original densities for outcomes in A by P (A). (Checking that this works

is left as an exercise for the reader).

For an event A with P (A) > 0, how should we define "the probability of B given that

A has occurred"? Obviously, no element of B outside A has occurred, so what we really

seek is the probability of B n A given that A has occurred. We will define this to be the

sum of the revised probability density over the elements of B n A. In other words, we

take P 1 (B n A) as the new, revised probability for B. The probability P 1 (B n A) can be

rewritten as

PI(B n A) - PI(w)

coEBflA

= P(A) p(o)

PB (lA)

I - ýp (Wo)

P(A)

P(B n A)

P (A)

This discussion motivates the following definition.

Definition 3. The conditional probability P (B I A) of B given that A has occurred is
defined by

P(B fl A)

P(B I A)- P(A)P (A)

where P (A) > 0.

Example 3. Define an experiment of rolling two fair dice and recording the total number
of pips on the top faces. Find the probability that the total number of pips is nine given that
the first die shows five pips on its top face.

Solution. P(total 9lfirst die 5) = P(total 9 and first die 5)/P(first die 5). Assume the
uniform probability density on the 36 pairs of possible outcomes for the two fair dice.
Define A = {(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)} and B = {(3, 6), (4, 5), (5, 4), (6, 3)1.
Then, we can write the probability as

P(BIA) = P(B n A)/P(A) = P({(5, 4)})/P(A)=(1) / (6) = 1/6. U

To relate the notion of conditional probability with the notion of independence from
Section 8.5.1, let us see what happens to the conditional probability of B given the occur-
rence of A if A and B are independent events:

P(B I A) =P(B n A)

P(A)

P(A) .P(B)

P(A)

- P(B)