Probability Calculator (Outcomes, Two Events, Repeated Trials and Normal Distribution)
Work out a probability with the steps shown: favorable ÷ total outcomes with odds, two events (and, or, neither, conditional), repeated trials with the binomial distribution, and the area under a normal curve.
How it works
P = favorable ÷ total · odds = favorable : unfavorable · independent: P(A and B) = P(A) × P(B) · P(A or B) = P(A) + P(B) − P(A and B) · P(A given B) = P(A and B) ÷ P(B) · binomial: P(X = k) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ · normal: P(x₁ < X < x₂) = Φ((x₂ − μ) ÷ σ) − Φ((x₁ − μ) ÷ σ)
Classical probability divides the outcomes you want by all equally likely outcomes, giving a number from 0 (impossible) to 1 (certain); odds express the same fact as favorable against unfavorable outcomes, and the calculator also gives the complement and the chance of seeing the event at least once in several tries. For two events, choose whether they are independent (one does not affect the other), mutually exclusive (they can’t happen together) or linked through a known P(A and B); you get the chance of both, at least one, exactly one, neither, each complement and both conditional probabilities, laid out in a joint probability table. Repeated trials use the binomial distribution — a fixed number of independent tries with the same chance of success — to give the probability of exactly, at most, at least, fewer or more than k successes, with the whole distribution charted. The normal option turns two values into z-scores and reads the area under the bell curve between them. Probabilities can be typed as decimals or percentages with the unit menu next to each field.
Worked example
Drawing one of the 3 red marbles from a bag of 8: P = 3 ÷ 8 = 0.375 = 37.5%, the odds in favor are 3 : 5, and drawing with replacement 10 times gives at least one red with probability 1 − 0.625¹⁰ ≈ 99.09%. For two independent events with P(A) = 0.5 and P(B) = 0.4, both happen with probability 0.2, at least one with 0.5 + 0.4 − 0.2 = 0.7 and neither with 0.3. Flipping a fair coin 10 times, exactly 5 heads has probability C(10, 5) × 0.5¹⁰ = 252 ÷ 1,024 ≈ 24.61%. For heights with mean 68 and standard deviation 4, the share between 60 and 72 is Φ(1) − Φ(−2) = 0.8413 − 0.0228 ≈ 81.86%.
Sources: NIST/SEMATECH e-Handbook — binomial distribution · NIST/SEMATECH e-Handbook — normal distribution
Frequently asked questions
What’s the difference between probability and odds?
Probability compares wins to everything (3 out of 8); odds compare wins to losses (3 to 5). Convert with P = for ÷ (for + against) — 3 : 5 in favor means 3 ÷ 8 = 37.5%.
How do I combine probabilities of two events?
For independent events, multiply for "both" (P(A) × P(B)); for "at least one", add them and subtract the overlap, P(A) + P(B) − P(A and B). Mutually exclusive events have no overlap, so "either" is simply P(A) + P(B). Two dice both showing six: 1/6 × 1/6 = 1/36.
What is the chance of something happening at least once?
Take the chance it never happens and subtract from 1: 1 − (1 − p)ⁿ. A 10% chance per try gives 1 − 0.9¹⁰ ≈ 65% over 10 tries — not 100%, because the tries can fail repeatedly.
When should I use the binomial option?
Use it when there is a fixed number of independent trials, each with only two outcomes (success or failure) and the same probability every time — coin flips, free throws, defective parts in a batch. It gives the chance of exactly, at most or at least k successes.
Why doesn’t a 50% chance mean alternating wins and losses?
Each independent try is fresh — probability describes the long-run average, not a pattern. Ten heads in a row leaves the next flip at exactly 50%; expecting otherwise is the gambler’s fallacy.
What does P(A given B) mean?
It is the chance of A once you already know B happened: P(A and B) ÷ P(B). If 20% of people are both left-handed and tall and 40% are tall, then 0.2 ÷ 0.4 = 50% of tall people are left-handed.
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This calculator is for educational purposes. Double-check important results before acting on them.