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Probability of Two Events Calculator

Combines the probabilities of two events A and B that are either independent or mutually exclusive into the probabilities of both, at least one, exactly one, one but not the other, and neither occurring.

When to use

You know P(A) and P(B) and how the events relate (independent or mutually exclusive) and need P(A and B), P(A or B), P(neither) or P(exactly one).

Do not use when: The events are dependent with a known conditional probability (apply P(A and B) = P(A)·P(B|A) directly), you need the probability of k successes in repeated trials (use binomial-distribution), or you need an expected payoff (use expected-value).

Formula

Independent: P(A and B) = P(A)·P(B). Mutually exclusive: P(A and B) = 0. Then P(A or B) = P(A) + P(B) − P(A and B); P(A not B) = P(A) − P(A and B); P(exactly one) = P(A) + P(B) − 2·P(A and B); P(neither) = 1 − P(A or B)

The identities follow from the addition rule of probability; independence is an assumption about the events, not something the calculator can check. Mutually exclusive events require P(A) + P(B) ≤ 1.

Inputs

Parameter	Type	Required	Description
`probability_a`	number	yes	Probability of event A, as a number between 0 and 1 (0.25 for 25 %). Range: ≥ 0, ≤ 1
`probability_b`	number	yes	Probability of event B, between 0 and 1. Range: ≥ 0, ≤ 1
`relationship`	enum: independent \| mutually_exclusive	default independent	How A and B relate: independent events multiply (P(A and B) = P(A)·P(B)); mutually exclusive events cannot occur together (P(A and B) = 0).

Outputs

Output	Type	Description
`probability_a_and_b`	number	Both events occur.
`probability_a_or_b`	number	At least one of the events occurs.
`probability_a_not_b`	number	A occurs but B does not.
`probability_b_not_a`	number	B occurs but A does not.
`probability_exactly_one`	number	Exactly one of the two events occurs.
`probability_neither`	number	Neither event occurs = 1 − P(A or B).

Example

Two independent coin flips (heads): {"probability_a":0.5,"probability_b":0.5,"relationship":"independent"} → {"probability_a_and_b":0.25,"probability_a_or_b":0.75,"probability_a_not_b":0.25,"probability_b_not_a":0.25,"probability_exactly_one":0.5,"probability_neither":0.25}

Mutually exclusive 0.2 and 0.3: {"probability_a":0.2,"probability_b":0.3,"relationship":"mutually_exclusive"} → {"probability_a_and_b":0,"probability_a_or_b":0.5,"probability_a_not_b":0.2,"probability_exactly_one":0.5,"probability_neither":0.5}

GET https://tttkmbb.com/api/v1/calculate/probability-of-events?probability_a=0.5&probability_b=0.5&relationship=independent

Machine access

API: GET https://tttkmbb.com/api/v1/calculate/probability-of-events (query parameters) or POST with a JSON body {"inputs": {...}}
Schema: https://tttkmbb.com/api/v1/calculators/probability-of-events · Markdown: https://tttkmbb.com/statistics/probability-of-events.md · JSON definition: https://tttkmbb.com/statistics/probability-of-events.json
MCP: server https://tttkmbb.com/mcp, tool run_calculator with calculator_id="probability-of-events"
OpenAPI operationId: calculate_event_probabilities
Freshness: static. Authentication: none. Rate limit: fair use (see rate limits).

Sources

Wikipedia – Independence (probability theory) (reference)
Wikipedia – Mutual exclusivity (reference)

FAQ

What is the difference between independent and mutually exclusive?

Independent events do not influence each other (rolling a die twice); mutually exclusive events cannot happen together (rolling a 1 and rolling a 6 on the same roll). Two events with non-zero probability cannot be both.

Can I enter percentages?

No, enter fractions between 0 and 1: 0.3 for 30 %. The outputs are fractions as well.

Related calculators

Binomial Distribution Calculator — Probability of k successes in repeated independent trials.
Expected Value Calculator — Average outcome of a probabilistic payoff.