This probability calculator answers three questions: how likely something is to happen at least once, how likely exactly k times, and the chance of two separate events together. Start with a single number on a European roulette wheel: a 1 in 37 chance, 2.7%. Over 37 spins, the calculator shows it lands at least once only 63.7% of the time. It’s free, and it shows the maths underneath.
Probability calculator: three questions it answers
At least once in n tries
The trick is to work out the chance it never happens and take that away from 1. A six on a die is 1 in 6. The chance of no six in four rolls is (5/6)⁴ = 48.2%, so the chance of at least one six is 51.8%. That’s a famous gambling problem: in the 1650s the Chevalier de Méré won money betting on a six in four rolls, then lost it betting on a double six in 24 rolls of two dice. The second bet comes in only 49.1% of the time. Blaise Pascal worked out why.
Exactly k times in n tries
This is the binomial formula. Roll a die six times and the chance of exactly one six is 40.2%. You “expect” one six, and it still happens less than half the time. Exactly two sixes: 20.1%. None at all: 33.5%.
A and B, A or B
For independent events, multiply for “both” and use P(A) + P(B) − P(A and B) for “either”. A 50% chance and a 30% chance both happening is 15%. At least one of them: 65%.
Where people go wrong with probability
- Adding instead of multiplying. Two 50% events aren’t 100% likely to give at least one success. It’s 75%.
- Thinking “due”. Red has come up eight times in a row? The next spin is still 18 in 37. The wheel doesn’t remember. Our roulette calculator lists the odds for every bet.
- Treating linked events as independent. Two horses in the same race can’t both win, and a team’s win and its clean sheet tend to come together. The multiplication rule only works when one result doesn’t change the other.
For totals with several dice, like rolling 7 with two, the dice probability calculator lists every outcome. To turn bookmaker odds into a percentage, use the implied probability calculator.