Conditional probability

The essentials

The conditional probability of A given B, written P(A|B), is the probability that A occurs when we already know that B has occurred. Knowing B shrinks the sample space to B, so only the outcomes of A that lie inside B count: P(A|B) = P(A ∩ B)/P(B), provided P(B) ≠ 0.

For equally likely outcomes, P(A|B) = n(A ∩ B)/n(B). Example: a die is thrown and the number is known to be even. The reduced sample space is {2, 4, 6}, so the probability that it is 6 is 1/3, not 1/6.

Key points

• 0 ≤ P(A|B) ≤ 1 and P(S|B) = P(B|B) = 1.
• P(A′|B) = 1 − P(A|B).
• P(A ∪ C | B) = P(A|B) + P(C|B) − P(A ∩ C | B).
• In general P(A|B) and P(B|A) are different.
• Use P(A ∩ B) = P(A) + P(B) − P(A ∪ B) when the union is given.

Exam tips

• List the reduced sample space when outcomes are few; it avoids formula errors.
• Read the question carefully to decide which event is the condition (the "given" part).

Common mistakes

• Dividing by P(A) instead of P(B) when finding P(A|B).
• Forgetting that "at least one head" excludes only TT, so it has three outcomes in two tosses.