Probability Calculator

How it works

A probability is a number from 0 (impossible) to 1 (certain). When every outcome is equally likely, the probability of an event is simply the number of outcomes that count as the event divided by the total: one face of a die is 1 ÷ 6. The complement, the chance the event does not happen, is 1 minus the probability.

For two events there are two rules. If the events are independent (one happening tells you nothing about the other, like two separate coin flips) the chance of both is the product, P(A) × P(B). The chance of at least one of them is P(A) + P(B) − P(A) × P(B): adding the two counts the overlap twice, so it is subtracted once. If the events are mutually exclusive (they cannot both happen, like a die showing 1 and showing 2) there is no overlap, so P(A or B) is just P(A) + P(B).

"At least one success in n tries" is easiest through its complement: the only way to get no successes is to fail every time, which has probability (1 − p)ⁿ, so the answer is 1 − (1 − p)ⁿ. This is why rare events become likely given enough attempts.

Formula

P(A) = favorable ÷ total           P(not A) = 1 − P(A)
Independent:        P(A and B) = P(A) × P(B)
                    P(A or B)  = P(A) + P(B) − P(A) × P(B)
Mutually exclusive: P(A or B)  = P(A) + P(B)        P(A and B) = 0
At least one in n:  P = 1 − (1 − p)ⁿ

Example

Rolling a 6 on one die: 1 favorable outcome out of 6, so P = 1 ÷ 6 = 0.166667 (16.67%) and the chance of not rolling a 6 is 0.833333.

Two independent events with P(A) = 0.5 and P(B) = 0.3: P(A and B) = 0.5 × 0.3 = 0.15, and P(A or B) = 0.5 + 0.3 − 0.15 = 0.65. If instead they were mutually exclusive, P(A or B) would be 0.5 + 0.3 = 0.8.

At least one success in 10 trials with p = 0.1 each: 1 − 0.9¹⁰ = 1 − 0.348678 = 0.651322, about 65%. The classic version is at least one six in four rolls of a die: 1 − (5/6)⁴ = 0.517747, just better than even.

Assumptions and limitations

• "Independent" means one event happening tells you nothing about the other: separate flips, separate draws with replacement. Drawing cards without replacement is not independent, and the product rule is wrong for it.
• "Mutually exclusive" means the events cannot both happen, so their probabilities cannot add to more than 1.
• The at-least-one formula assumes every trial has the same probability and the trials are independent of each other.
• Probabilities are entered as decimals between 0 and 1, not as percentages.

Frequently asked questions