P(A∩B) is the probability that both event A and event B occur at the same time. It is also called the joint probability of A and B.

How is P(A∩B) calculated for independent events?

For independent events, P(A∩B) = P(A) × P(B). Independence means the occurrence of one event does not affect the other.

What does mutually exclusive mean?

Mutually exclusive events cannot both happen at the same time. For example, rolling a 1 and rolling a 6 on a single die roll. Their intersection P(A∩B) is always 0.

When do I use Custom (conditional) mode?

Use Custom mode when you know P(A|B) — the probability of A given B. The formula is P(A∩B) = P(A|B) × P(B). This handles correlated events that are neither independent nor mutually exclusive.

Why does the calculator show an error about P(A∩B) > min(P(A), P(B))?

A joint probability can never exceed either individual probability. If your inputs violate this rule, the values are inconsistent — check that P(A|B) ≤ 1.

What is the complement of P(A∩B)?

The complement is P((A∩B)') = 1 − P(A∩B), representing the probability that it is NOT the case that both A and B occur simultaneously.

Probability Intersection Calculator

Name: Probability Intersection Calculator
Availability: InStock
Author: Nham Vu

Calculate P(A∩B) for independent events, mutually exclusive events, or enter a known joint probability directly.

Event Probabilities

P(A) — Probability of Event A

P(B) — Probability of Event B

Calculation Mode

Independent Events

P(A∩B) = P(A) × P(B)

Mutually Exclusive

P(A∩B) = 0 — events cannot co-occur

Custom (Conditional)

P(A∩B) = P(A|B) × P(B)

Enter probabilities and click Calculate

Summary

Calculate P(A∩B) for independent events, mutually exclusive events, or enter a known joint probability directly.

How it works

Enter P(A) and P(B) as decimals (e.g. 0.5) or percentages (e.g. 50%).
Select a calculation mode: Independent, Mutually Exclusive, or Custom (conditional).
For Custom mode, enter P(A|B) — the probability of A given B has occurred.
Click Calculate to compute P(A∩B) along with complementary probabilities.
Use the Venn diagram preview to visualize the overlap between the two events.

Use cases

Determine the probability that two independent dice rolls both land on a specific face.
Verify that two events are mutually exclusive and their intersection is zero.
Calculate the joint probability of two correlated market events using historical conditional data.
Solve homework problems involving basic probability and set theory.
Model the likelihood of two independent hardware failures occurring simultaneously.
Check probability rules and validate that P(A∩B) ≤ min(P(A), P(B)).

Frequently Asked Questions

Last updated: 2026-06-13 · Reviewed by Nham Vu