Conditional probability (P(A|B))

How to read it

OpenStax’s Introductory Statistics defines P(A|B) as the probability that A will occur given that B has already occurred, and explains that the condition shrinks the sample space: only outcomes in B are left to count. Its formula is P(A|B) = P(A AND B) / P(B). Grinstead and Snell’s Introduction to Probability (Dartmouth) gives a die example. Roll once; the chance of a 6 is 1/6. Told that the roll was more than 4, only 5 and 6 remain, and the probability of a 6 becomes 1/2: P(6 | more than 4) = 1/2. OpenStax’s Contemporary Mathematics calls it the updated probability of an event once something is assumed to have occurred.

Also searched as: given symbol, P(A|B), P(A given B), P(B|A), probability of A given B, vertical bar in probability.

P(A|B) and P(B|A)

The order matters. P(A|B) and P(B|A) are different questions: OpenStax’s Contemporary Mathematics sets them as separate exercises, the chance that a student is a sophomore given that they study the arts and the chance that a student studies the arts given that they are a sophomore, and its Introductory Statistics asks readers to work out both and check whether they are equal.

Look-alikes

The bar is the same character as the “such that” bar in set-builder notation, but in probability it is read “given”. It is not a division sign, though Keynes’s older p/q notation used a slash.