Table of Contents
What Is Joint Probability?
Joint probability is the probability that two events A and B both occur simultaneously. It is denoted P(A ∩ B) or P(A and B). Joint probability is fundamental in probability theory and is used to analyze the relationship between events.
For independent events, the joint probability is simply the product of individual probabilities. For dependent events, you must account for the conditional probability of one event given the other.
Formulas
Examples
| Scenario | P(A) | P(B) | P(A∩B) |
|---|---|---|---|
| Two coins both heads | 0.5 | 0.5 | 0.25 |
| Rain and umbrella | 0.3 | 0.6 | Depends on P(B|A) |
| Two dice both 6 | 1/6 | 1/6 | 1/36 |
Frequently Asked Questions
What is the difference between joint and marginal probability?
Marginal probability is the probability of a single event (e.g., P(A)), regardless of other events. Joint probability considers two or more events occurring together. Marginal probabilities can be obtained by summing joint probabilities across all values of the other variable.
How do I know if events are independent?
Events A and B are independent if P(A ∩ B) = P(A) × P(B), or equivalently, if P(A|B) = P(A). If knowing B occurred changes the probability of A, they are dependent.
Can joint probability be greater than marginal probability?
No, joint probability can never exceed either marginal probability. P(A ∩ B) ≤ min(P(A), P(B)). The intersection of two sets is always contained within each individual set.