The conditional probability P(A given B)

The idea: information that changes things

The conditional probability of A given B, written P(A|B), represents the probability that event A occurs, knowing that we already know event B has occurred. Knowing that B has taken place restricts the universe of possibilities to B, which generally changes the probability of A.

Formula

If P(B) is not 0, we define:

P(A given B) = P(A intersection B) / P(B)

(A intersection B denotes the event "A and B both occur")
(this division only makes sense if P(B) is strictly positive)

We immediately deduce the multiplication rule, very useful in practice:

P(A intersection B) = P(B) x P(A given B)

Numerical example

In a group, 40 percent of people play a sport (event B), and among them, 25 percent also play a musical instrument (event A given B). The probability that a person picked at random plays both a sport and an instrument is:

P(A intersection B) = P(B) x P(A|B) = 0.40 x 0.25 = 0.10, i.e. 10 percent of the group.

Common pitfall

Never confuse P(A|B) with P(B|A): these are two generally different quantities. For example, the probability of being ill given that the test is positive is not the same thing as the probability that the test is positive given that one is ill (see the lesson on the medical test). Confusing the two is the most frequent mistake in this chapter.