Joint probability

Joint probability is the chance that two events occur together, P(A and B), read from the inside cells of a contingency table rather than its row or column totals.

ByHoang TruongUpdated July 11, 2026

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A retailer cross-tabulates 200 customers by membership and purchase: 40 members purchased, 60 members did not, 30 non-members purchased, and 70 non-members did not. The joint probability that a customer is both a member and a purchaser is the single cell divided by the grand total: 40 ÷ 200 = 20%, distinct from the 50% marginal probability of membership or the 40% conditional probability of purchasing given membership.

Where it fits

TopicProbability & DistributionsCoreSubjectData Analysis & StatisticsCore

The formula

LaTeX

P(A \cap B) = \frac{f_{AB}}{N}

Variables

$f_{A B}$: Count of observations in the (A and B) cell
$N$: Grand total number of observations

Gives the joint probability of two events directly from a contingency table.

LaTeX

P(A \cap B) = P(A\mid B)\,P(B)

Variables

$P (A \cap B)$: Joint probability of A and B occurring together
$P (A ∣ B)$: Conditional probability of A given B
$P (B)$: Marginal probability of B

Relates a joint probability to a conditional probability and a marginal probability.

Check yourself

PracticeCORE

A survey of 150 employees cross-tabulates department (Sales / Operations) against satisfaction (Satisfied / Not satisfied): Sales-Satisfied = 45, Sales-Not satisfied = 15, Operations-Satisfied = 40, Operations-Not satisfied = 50. What is the joint probability that a randomly chosen employee works in Operations AND is satisfied?

Select an answer to check your understanding.