Probability Calculator
Basic probability, complements, combinations (nCr), permutations (nPr), union and intersection of events. Customize mode adds conditional probability, Bayes' theorem, dependent events, symmetric difference, and expected value.
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How to use
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Pick the calculation you need.
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Fill in the inputs that appear for that mode.
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Probabilities can be entered as decimals between 0 and 1 (e.g. 0.25 = 25%).
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Switch to Customize for conditional probability, Bayes' theorem, dependent events, symmetric difference, and expected value.
Frequently asked questions
Combinations (nCr) count selections where order doesn't matter — picking 3 students from 10 for a group photo. Permutations (nPr) count arrangements where order matters — picking 3 students for gold, silver, bronze medals. nPr is always larger than or equal to nCr.
Two events are independent if one happening doesn't affect the probability of the other. Coin flips and dice rolls are independent. Drawing cards without replacement is NOT independent. The Quick union/intersection formulas assume independence; switch to dependent modes in Customize when P(A∩B) is known.
Bayes' theorem flips conditional probabilities. Given the PRIOR P(A), the LIKELIHOOD P(B|A), and the FALSE POSITIVE rate P(B|¬A), you can compute the POSTERIOR P(A|B). Classic example: 1% of a population has a disease, test is 99% sensitive, 5% false positive. If you test positive, your actual probability of having the disease is only about 16.7% — not 99%. Bayes' theorem catches the base-rate fallacy.
P(A|B) is the probability of A given that B has already happened. Formula: P(A|B) = P(A ∩ B) / P(B). For independent events P(A|B) = P(A) — knowing B doesn't change the probability of A.
Expected value E[X] is the long-run average of a random variable, weighted by each outcome's probability. For a fair coin paying $1 for heads and $0 for tails: E[X] = 1·0.5 + 0·0.5 = $0.50. It's the average you'd converge to over many trials. Used in finance, insurance, and game theory.
JavaScript's Number type overflows around 170! (≈ 7.3 × 10³⁰⁶). For factorials and combinations involving larger n, the calculator would return Infinity, which is useless. If you need bigger values, use a symbolic math system.
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Source: Standard probability formulas · Last verified 2026-06. This tool provides estimates only and is not legal, tax or financial advice. Always verify your specific situation with the relevant UAE authority or a licensed advisor before taking action.