Probability Calculator

A number between 0 and 1 saying how often we expect an event to happen. 0 = never, 1 = always, 0.5 = half the time. We compute it by dividing favorable outcomes by total outcomes (when each outcome is equally likely) — flipping a coin to get heads is 1 favorable ÷ 2 total = 0.5.

No — and this is the most common misconception. P = 0.25 is a LONG-RUN average. In any small batch (say 20 attempts) it might happen 0 times, or 5 times, or 10. The actual rate converges to 25% only over many, many trials. "1 in 4" is a frequency, not a schedule. A 1-in-100 risk per launch can hit on the very first launch — or never in 500.

Only when A and B are independent — when one event doesn't change the probability of the other. Two coin flips are independent. Two dice rolls are independent. Drawing two cards WITHOUT putting the first back is NOT independent (the deck shrinks). For dependent events you need the conditional formula P(A and B) = P(A) × P(B|A).

Because the case where BOTH A and B happen is counted once by P(A) and a second time by P(B) — adding them double-counts it. We subtract the overlap once to fix the count. This is the inclusion-exclusion principle in its simplest form. For mutually exclusive events (where P(A and B) = 0), the formula collapses to P(A) + P(B).

Instead of summing many cases ("the event happens on trial 1, OR trial 2, OR trial 3, …"), compute the probability it NEVER happens — that's a single product: (1 − p)^n. Then subtract from 1. Example: at least one six in 4 dice rolls = 1 − (5/6)^4 ≈ 51.8%. The complement trick is faster and easier than enumerating cases.

Whenever you want the probability of EXACTLY k successes (or at-most-k, or at-least-k) in n independent trials, each with the same success probability p. Examples: exactly 3 heads in 5 flips, at most 2 defects in 10 items, at least 7 correct on a 10-question quiz. Formula: P(X = k) = C(n, k) × p^k × (1 − p)^(n − k). Sum these terms across a range for the at-most/at-least cumulative.

"The probability of A GIVEN that B happened" — written P(A | B). Read the bar (|) as "given." Formula: P(A | B) = P(A and B) ÷ P(B). It restricts attention to the slice of world where B is already true and asks what fraction of THAT slice also has A. Example: P(face card | red card) — given a red card was drawn, what's the probability it's a face card?

Bayes' theorem flips a known forward conditional. If you know P(B | A) (e.g., test sensitivity: 99% of sick people test positive), the prior P(A) (1% of people are sick), and the total P(B) (overall positive rate), Bayes gives you the inverse P(A | B): given a positive test, what's the actual probability of being sick? The famous answer for these inputs is ~17% — much lower than people intuitively expect. Critical for medical diagnosis, spam filters, criminal forensics, anywhere you update a belief after evidence.

It's friendlier as long as you remember it's a long-run average. P = 0.04 ≈ "1 in 25" — the event happens roughly once per 25 trials on average. But "average" doesn't guarantee timing — see the misconception above. Use it for intuition, not for prediction.

Use the dedicated calculators. • Combinations C(n, r) — for unordered selections (poker hands, lottery). • Permutations P(n, r) — for ordered arrangements (race medals, PIN codes). • Factorial n! — for arranging all items in a row. Compute the count there, then plug favorable / total here.