Two independent events
Repeating event A
For independent events (one does not affect the other), P(A and B) = P(A) x P(B). P(A or B) = P(A) + P(B) - P(A and B), because the overlap would otherwise be counted twice. P(not A) = 1 - P(A), and P(neither) = (1 - P(A)) x (1 - P(B)). For repeated tries of the same event, the chance that it never happens is (1 - p)^n, so the chance of at least one success is 1 - (1 - p)^n. The chance of exactly k successes in n tries is the binomial formula: C(n, k) x p^k x (1 - p)^(n - k), where C(n, k) = n! / (k! x (n - k)!) counts the ways to choose which tries succeed.
| Event | Chance | Repeated or combined |
|---|---|---|
| Heads on one coin flip | 50% | Two heads in a row: 25%; at least one head in 3 flips: 87.5% |
| Rolling a six on one die | 16.67% | At least one six in 6 rolls: 66.51% |
| Drawing a heart from a deck | 25% | At least one heart in 3 draws with replacement: 57.81% |
| Winning a 1-in-10 raffle | 10% | Winning at least once in 10 raffles: 65.13% |
| Rain on any day with a 30% chance | 30% | Staying dry all 5 days: 16.81% |
"A and B" means both happen, which is rarer, so you multiply the chances (for independent events). "A or B" means at least one happens, which is more likely, so you add and subtract the overlap.
Two events are independent if one happening does not change the chance of the other, like two separate coin flips. Drawing cards without putting them back is not independent, so this calculator does not apply to that case.
Work out the chance of zero successes, (1 - p)^n, then subtract it from 1. For example, the chance of at least one six in 6 dice rolls is 1 - (5/6)^6 = 66.5%, not 100%.
Odds in favour = p / (1 - p). A 25% chance is 0.25 / 0.75 = 1 to 3 in favour, often written 1:3, or 3 to 1 against.
Each try is separate: the chance of at least one success in 2 tries of a 50% event is 1 - 0.5^2 = 75%. The chance only approaches 100% as the number of tries grows, and never quite reaches it.
Assumes independent events and a constant chance on every try. Results are rounded.
See also: factorial, permutation and combination calculator · standard deviation calculator · z-score calculator · percentage calculator · dice roller · coin flip · all tools