What is the difference between P(A AND B) and P(A OR B)?

P(A AND B) is the probability that BOTH events occur simultaneously; for independent events it equals P(A) × P(B). P(A OR B) is the probability that AT LEAST ONE event occurs; it equals P(A) + P(B) − P(A AND B).

Does this assume the events are independent?

Yes. The combined AND/OR formulas apply the standard multiplication and addition rules for independent events. If your events are dependent, the results will not be accurate.

What is complementary probability?

The complementary probability of event A, written P(A'), is 1 − P(A). It represents the probability that event A does NOT occur. For example, if P(A) = 0.3, then P(A') = 0.7.

How are fractions simplified?

The calculator divides both the numerator and denominator by their greatest common divisor (GCD) to produce the simplest equivalent fraction.

Can favorable outcomes exceed total outcomes?

No. Favorable outcomes must be between 0 and the total number of outcomes. The calculator will show an error if this condition is violated.

What does a probability of 0 or 1 mean?

A probability of 0 means the event is impossible (it can never occur), while a probability of 1 means the event is certain (it always occurs).

Probability Calculator

Name: Probability Calculator
Availability: InStock
Author: Nham Vu

Calculate the probability of single events, complementary events, and combined AND/OR probabilities for two independent events.

A Event A

Favorable outcomes

Total outcomes

Example: rolling a 3 on a standard die → favorable: 1, total: 6

B Event B (optional — for AND / OR)

Favorable outcomes

Total outcomes

Example: flipping heads on a coin → favorable: 1, total: 2

Enter outcomes above and click Calculate to see results.

Summary

Calculate the probability of single events, complementary events, and combined AND/OR probabilities for two independent events.

How it works

Enter the number of favorable outcomes and total possible outcomes for Event A.
Optionally enter values for Event B to enable combined probability calculations.
The calculator instantly computes P(A), P(A's complement), P(B), P(B's complement).
Combined results show P(A AND B) — both events occur — and P(A OR B) — at least one occurs.
All results display as decimals, percentages, and simplified fractions.

Use cases

Determine the probability of rolling a specific number on a die.
Calculate the chance of drawing a certain card from a deck.
Find the joint probability of two independent coin flips.
Compute the likelihood that at least one of two independent events occurs.
Cross-check probability homework answers step by step.
Understand complementary events (the probability something does NOT happen).
Calculate the AND probability for quality control scenarios.
Teach or learn the addition rule and multiplication rule of probability.

Frequently Asked Questions

Last updated: 2026-06-11 · Reviewed by Nham Vu