What is Probability Calculator?

Solve probability problems with built-in calculators for combinations (nCr), permutations (nPr), conditional probability, Bayes' theorem, binomial and normal distributions, and compound events. Enter values and see step-by-step solutions, useful for statistics coursework and data analysis.

Nine modes cover the everyday probability questions: single event (m / n), two events with the full breakdown (intersection, union, neither, exactly-one, complements), binomial (exactly k successes in n trials, plus at-most, at-least, mean and standard deviation), normal distribution (Z-score and the area below, above, or between values), at-least-once across a series of trials, combinations (n choose r, order doesn't matter), permutations (n permute r, order matters), conditional P(A given B), and Bayes' theorem for inverting conditionals. Factorials use a gamma-function approach up to n = 170; above that, double-precision floats overflow to infinity. Results update live as you type and show the formula, substitution, fraction form, decimal, and percentage so you can copy the working straight into homework.

How to use

1. Choose a calculator mode: single event probability, combinations/permutations, conditional probability, or Bayes' theorem.
2. Enter the required values (favorable outcomes, total outcomes, n, r, or prior/likelihood values).
3. View the result with a step-by-step formula breakdown, and copy or export the solution.

When to use

• Working through a stats homework problem and wanting to see the formula expanded step by step.
• Calculating poker hand odds — combinations of 5 cards drawn from a 52-card deck.
• Updating a prior belief with new evidence using Bayes' theorem (medical testing, spam filtering).

Result

How many ways can you pick 5 cards from 52? C(52,5) = 2,598,960. What's the probability of all 5 being hearts? C(13,5)/C(52,5) = 0.000495 or about 1 in 2,019.

FAQ

What is the difference between combinations and permutations?

Combinations count unordered selections — picking apple, banana, orange is the same as orange, apple, banana. Permutations count ordered ones — first, second, and third place medals are distinct. C(n,r) = n! / (r!(n-r)!), and P(n,r) = n! / (n-r)! is always r! times larger.

Why does the calculator cap n at 170?

170! is about 7.26 × 10^306, the largest factorial that fits in a double-precision float. 171! overflows to Infinity, so combinations and permutations stop returning useful numbers above that. For larger inputs, switch to a logarithmic or BigInt approach.

Can I use this for dependent events?

Yes. The conditional probability mode handles P(A|B) = P(A∩B)/P(B), which is the standard definition of dependent events. For chains of dependent events multiply conditionals step by step, e.g. drawing two aces without replacement is 4/52 × 3/51.

What does Bayes' theorem actually let me calculate?

It flips a conditional. If you know P(test positive | disease) plus the disease rate, Bayes gives you P(disease | test positive) — the direction you actually care about. The classic worked example shows why a 99% accurate test can still produce mostly false positives for a rare disease.

Are probabilities always between 0 and 1?

Yes. Zero means impossible, one means certain. Anything outside that range signals an input error. The calculator flags impossible inputs (favorable > total, r > n, P(A∩B) > P(B)) before computing so you don't get nonsense numbers.

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