Probability Calculator

Single-event probability, plus combined probability of independent events happening (AND) or at least one happening (OR)

Combining probabilities of multiple events — whether both happen (AND) or at least one happens (OR) — uses different combination rules depending on which question is actually being asked, and whether the events are independent, which is exactly why "combined probability" isn't a single formula but a family of related calculations.

Why AND and OR probability combine differently

For independent events, the probability that both occur (AND) is calculated by multiplying the individual probabilities together — since each event's outcome doesn't affect the other, and requiring both to happen is a more restrictive (lower-probability) condition than requiring just one. The probability that at least one of the events occurs (OR) is calculated differently — for independent events, it's often easier to calculate as one minus the probability that none of the events occur, rather than directly summing individual probabilities (which would double-count scenarios where multiple events occur together, if calculated naively as simple addition). These are structurally different calculations because "both happen" and "at least one happens" are asking about fundamentally different, non-equivalent combinations of the same underlying individual event probabilities.

A worked example

Two events with the same individual probability produce a lower combined probability for "both happen" (AND) than for "at least one happens" (OR) — requiring both events is always a more restrictive condition than requiring just one, so the AND probability is always less than or equal to the OR probability for the same set of underlying events, never the reverse.

How this connects to your other math and statistics calculations

Probability calculations often depend on permutation or combination counts as inputs — the Permutation and Combination Calculator handles the "how many ways can this happen" counting that frequently underlies the raw probability figures used in a broader probability calculation.

Common mistakes

Simply adding individual probabilities together to calculate the probability that at least one of several events occurs (rather than using the correct combination formula) overstates the true combined probability whenever there's any chance of multiple events occurring together — naive addition double-counts those overlapping scenarios, which is exactly the error the correct OR-combination formula is specifically designed to avoid.