How to use the probability calculator
- nCr / nPr: enter how many items there are and how many are chosen to get combinations, permutations and the odds of one specific pick (for example lottery odds).
- Two events: enter P(A) and P(B), and either tick "independent" or give P(A and B), to get unions, intersections, complements and conditional probabilities.
- Binomial: enter the number of trials, the success probability and k to get exact and cumulative probabilities.
- At least once: find the chance something happens at least once in repeated tries, and how many tries you need for a 50% or 95% chance.
The formulas
Combinations count selections where order does not matter: C(n, r) = n! ÷ (r!(n − r)!). Choosing 6 numbers from 49 gives 13,983,816 combinations, so one ticket has odds of 1 in nearly 14 million. Permutations count ordered arrangements: P(n, r) = n! ÷ (n − r)!. Results are computed with exact big-integer arithmetic, so even C(1000, 500) is exact.
For two events, P(A or B) = P(A) + P(B) − P(A and B), and for independent events P(A and B) = P(A) × P(B). Conditional probability is P(A | B) = P(A and B) ÷ P(B). The binomial distribution gives the probability of exactly k successes in n independent trials: C(n, k) pᵏ(1 − p)ⁿ⁻ᵏ. "At least once" uses the complement: 1 − (1 − p)ⁿ. For example, the chance of rolling at least one six in six throws of a die is 1 − (5/6)⁶ ≈ 66.5%, not 100%. Calculations run locally in your browser.
What is the difference between combinations and permutations?
Combinations ignore order (a lottery draw); permutations count different orders as different (a race podium).
How do I enter percentages?
Type them with a % sign, like 15%. Fractions like 1/6 also work.
Why isn't the chance of a six in six rolls 100%?
Because the rolls are independent: the chance of no six is (5/6)⁶ ≈ 33.5%.