Conditional probability

Conditional probability is the probability of event A given that event B has occurred, written P(A | B) = P(A ∩ B) / P(B). It adjusts the likelihood of A by restricting the sample space to B.

ByHoang TruongUpdated June 30, 2026

See it move

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Thirty per cent of a retailer's customers hold a loyalty card, and 18% both hold a card and made a purchase last month. Restricting attention to card holders — the conditioning event — gives P(purchase given card) = 18% ÷ 30% = 60%, well above the unconditional purchase rate. If that conditional probability had equalled the plain 30%, card ownership and purchasing would be independent.

Where it fits

TopicProbability & DistributionsCoreSubjectData Analysis & StatisticsCore

The formula

LaTeX

P(A \mid B) = \frac{P(A \cap B)}{P(B)}

Variables

$P (A ∣ B)$: Probability of event A given that event B has occurred
$P (A \cap B)$: Probability that both A and B occur
$P (B)$: Probability of event B; must be greater than zero

Events A and B are statistically independent if P(A | B) = P(A), meaning knowing B gives no information about A.

Check yourself

PracticeCORE

A firm has 500 sales leads on its database. Of these, 150 are classified as 'warm' (prior contact made). Records show that 90 of the 150 warm leads converted into sales last quarter. What is the probability that a lead converted into a sale, given that it is classified as warm?

Select an answer to check your understanding.