What Is Joint Probability?
Joint probability is the probability that two events A and B both happen, written P(A ∩ B). The way you compute it depends on whether the events are independent or dependent.
For independent events, where one event does not affect the other, the joint probability is simply the product of the two individual probabilities. For dependent events, you must use a conditional probability: the joint probability equals the conditional probability P(A|B) times the probability of the conditioning event P(B).
Formula
Independent events:
P(A ∩ B) = P(A) × P(B)
Dependent events:
P(A ∩ B) = P(A|B) × P(B)
where:
P(A), P(B) = individual event probabilities
P(A|B) = probability of A given B
Applications
- Finding the chance that two independent systems both fail at the same time
- Computing the probability of a customer belonging to two overlapping segments
- Estimating the chance that a patient has a disease and tests positive for it
- Building probability trees and risk models that combine multiple events
Sources
- Moore, D. S., McCabe, G. P., & Craig, B. A. (2021). Introduction to the Practice of Statistics (10th ed.). Macmillan.
- Montgomery, D. C., & Runger, G. C. (2018). Applied Statistics and Probability for Engineers (7th ed.). Wiley.