Probability Calculator

Frequently Asked Questions

What is the difference between permutation and combination?

Permutations count ordered arrangements: P(n,r) = n!/(n−r)!. Combinations count unordered selections: C(n,r) = n!/(r!(n−r)!). For 3 from 5: P = 60, C = 10.

How do I calculate probability of independent events?

Multiply: P(A and B) = P(A) × P(B). For two coin flips landing heads: 0.5 × 0.5 = 0.25.

What is the addition rule for mutually exclusive events?

P(A or B) = P(A) + P(B) when both can't happen at once. If they can overlap, subtract the intersection: P(A) + P(B) − P(A and B).

What does it mean if probability is 0 or 1?

Probability of 0 means impossible; 1 means certain. All real-world events fall between, often expressed as percentages from 0% to 100%.

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How This Calculator Works

This tool covers three common probability tasks. Single event computes the chance of an outcome as favorable outcomes divided by total outcomes, shown as a decimal, a percentage and an odds ratio. Combinations counts the number of ways to choose r items from n, both when order does not matter (combinations) and when it does (permutations). Multiple events takes two independent probabilities and derives the AND, OR and complement results. Pick a mode, enter the values, and the answers appear instantly.

Formula

Basic probability: P = favorable ÷ total, always between 0 and 1. Combinations: C(n,r) = n! ÷ (r! × (n − r)!); permutations: P(n,r) = n! ÷ (n − r)!. For independent events the multiplication rule gives P(A and B) = P(A) × P(B), the addition rule gives P(A or B) = P(A) + P(B) − P(A and B), and the complement rule gives P(not A) = 1 − P(A).

Worked Example

Rolling a die, the chance of a number greater than 4 (a 5 or 6) is P = 2 ÷ 6 = 0.333, or 33.3%, with odds of 2 : 4. Choosing 3 toppings from 10: C(10,3) = 10! ÷ (3! × 7!) = 120 combinations, while ordered selections give P(10,3) = 720. For two independent events with P(A) = 0.5 and P(B) = 0.3: P(A and B) = 0.15 and P(A or B) = 0.5 + 0.3 − 0.15 = 0.65.

Common Mistakes

Multiplying probabilities of dependent events - the AND rule shown assumes independence.

Forgetting to subtract the overlap in P(A or B) - omitting it double-counts shared outcomes.

Swapping combinations and permutations - use permutations only when order matters.

Entering more favorable than total outcomes - a probability can never exceed 1.

Using n above 20 in combinations - factorials grow huge and the tool caps n at 20.

Tips & FAQ

What is expected value? The long-run average, Σ(value × probability) - positive means a favorable bet, negative an unfavorable one. Why do casinos always win? Every game has a slightly negative expected value for the player. What is the law of large numbers? Over many trials the observed frequency converges to the true probability. How do odds relate to probability? Odds of a : b mean probability a ÷ (a + b).

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