conditional probability

Also called: P(A | B), probability given

The probability of A when you already know B is true, written P(A | B) and read "A given B": P(A and B) ÷ P(B). The condition shrinks the denominator to the cases where B holds.

A conditional probability answers "how likely is A among the cases where B is true?" Count only the B cases and see what share of them is A:

$P(A \mid B) = \frac{P(A \text{ and } B)}{P(B)}$

In a 2×2 table this means reading one row or one column instead of the whole table. Precision is P(really positive | flagged), the flagged row; recall is P(flagged | really positive), the positive column. Same top-left cell, different denominators.

The direction matters: P(A | B) and P(B | A) share a numerator but divide by different things, so they can be very different. A p-value is a probability of data this extreme given no real effect, not the probability that there is no effect.

Example

Klasna’s churn model flags 120 of 1,000 Club members; 48 of them cancel, and 80 cancel in total.

P(cancel) = 80 ÷ 1,000 = 0.08.
P(cancel | flagged) = 48 ÷ 120 = 0.40 (precision).
P(flagged | cancel) = 48 ÷ 80 = 0.60 (recall).

Common mistakes

Swapping the condition. P(A | B) ≠ P(B | A): 0.40 vs 0.60 above.
Reading a p-value backwards. It conditions on "no effect"; it is not P(no effect).
Trusting accuracy on rare events. A model that flags nobody scores 92% here and catches nothing.
Dividing by the whole table. The denominator of a conditional probability is the condition’s row or column.

Learn it in the course

Conditional probability and the 2×2 table · P(A | B) as a share inside a row, precision, recall and accuracy, relative risk and the odds ratio, and why the direction of a condition matters.