Probability Calculator

How to use the probability calculator

1. Choose a mode. Pick two independent events to combine two probabilities, or repeated trials for the same event tried several times.
2. Enter probabilities as percentages from 0 to 100. A fair coin is 50; rolling a six on one die is 16.67.
3. For repeated trials, enter the number of trials n (1 to 1,000) and the number of successes k you're asking about. k can't be greater than n.
4. Read the results. Two-event mode shows P(A and B), P(A or B), exactly one, neither and not A. Repeated-trials mode shows exactly k, at most k, at least k, at least one and the expected number of successes.

Formulas and assumptions

Both modes assume independence: the outcome of one event doesn't change the probability of the other. Coin flips, dice rolls and draws with replacement are independent; drawing cards without putting them back is not.

Two independent events

• Both: P(A and B) = P(A) × P(B)
• At least one: P(A or B) = P(A) + P(B) − P(A) × P(B)
• Exactly one: P(A) + P(B) − 2 × P(A) × P(B)
• Neither: (1 − P(A)) × (1 − P(B))
• Not A: 1 − P(A)

Repeated trials (binomial distribution)

For n independent trials, each with success probability p, the chance of exactly k successes is:

P(X = k) = C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ

where C(n, k) is the number of ways to choose k trials out of n. P(X ≤ k) adds up the terms from 0 to k, and P(X ≥ k) is 1 − P(X ≤ k) + P(X = k). The chance of at least one success is 1 − (1 − p)ⁿ, and the expected number of successes is n × p. The calculator works with logarithms of factorials, so large n stays accurate.

Examples

1. Two backup drives

Suppose each of two drives has a 3% chance of failing in a given year, independently of the other. The chance that both fail is 0.09%, the chance that at least one fails is 5.91% and the chance that neither fails is 94.09%. A second drive turns a 3% risk of losing everything into less than one in a thousand.

2. Sixes in ten rolls

Roll a fair die 10 times (p = 16.67%, n = 10, k = 2):

3. Free throws

A 75% free-throw shooter takes 10 shots. The chance of making exactly 8 is 28.16%, and the chance of making at least 8 is 52.56%. On average the shooter makes 7.5. For just two shots, two-event mode gives 56.25% for both, 37.5% for exactly one and 6.25% for missing both.

Reading the results

A probability is a long-run frequency, not a promise. A 29% chance of exactly two sixes means that in many sets of 10 rolls, roughly 29 out of 100 sets end that way. Over a single set, anything can happen.

Check the independence assumption before relying on the numbers. Weather on consecutive days, a team's games in a slump or parts from the same faulty batch tend to be linked, and multiplying their probabilities will understate how often bad outcomes cluster together.