Probability Calculator

Two kinds of probability question

A probability is a number from 0 (impossible) to 1 (certain), often written as a percentage from 0% to 100%. Choose how you want to type it — a probability of 0.25 and 25% are the same thing.

Two events combines the chances of two separate things, A and B. Enter P(A) and P(B) to see the chance that both happen, that at least one happens, that exactly one happens, that only A or only B happens, or that neither does. The table under the result lays all four combinations out as a grid.

Repeated trials is for doing the same thing several times — flipping a coin, rolling a die, sending out applications — where each attempt succeeds with the same probability p. Enter p, the number of trials n and a number of successes k to see the chance of exactly k, at least k, at most k, and at least one success. The chart shows the whole distribution.

The independence assumption

Both modes assume independence: the outcome of one event or trial doesn’t change the chances of another. Coin flips, dice rolls and draws where the card is put back are independent. Many real-world events are not: rain today makes rain tomorrow more likely, and drawing a card without putting it back changes what is left in the deck. For dependent events these formulas give the wrong answer, and you need a conditional probability instead (see below).

Probability formulas

For two independent events A and B:

For n independent trials with success probability p, the number of successes X follows the binomial distribution:

C(n, k)

the number of ways to choose which k of the n trials are successes: n! ÷ (k! × (n − k)!)

pᵏ × (1 − p)ⁿ⁻ᵏ

the chance of one particular order of k successes and n − k failures

P(X ≥ k)

the sum of P(X = i) for every i from k to n; P(X ≤ k) sums from 0 to k

The expected number of successes is n × p and the standard deviation is √(n × p × (1 − p)). Binomial coefficients and powers become astronomically large or small as n grows, so the calculator works with logarithms and adds the tail probabilities term by term. Results are accurate to about nine significant digits up to 100,000 trials, and probabilities far smaller than 10⁻³⁰⁰ are still reported.

Worked examples

Two job applications. You estimate a 60% chance of an interview at one company and 30% at an unrelated one, and treat them as independent.

1. Both: 0.6 × 0.3 = 0.18, so 18%.
2. At least one: 0.6 + 0.3 − 0.18 = 0.72, so 72%.
3. Neither: (1 − 0.6) × (1 − 0.3) = 0.4 × 0.7 = 0.28, so 28% — and 72% + 28% = 100%.
4. Exactly one: 0.6 + 0.3 − 2 × 0.18 = 0.54, so 54% (42% only the first, 12% only the second).

Ten coin flips. With p = 0.5, n = 10 and k = 5:

1. Ways to get 5 heads: C(10, 5) = 10! ÷ (5! × 5!) = 252.
2. Chance of any one order: 0.5⁵ × 0.5⁵ = 0.5¹⁰ = 1/1,024 ≈ 0.000977.
3. Exactly 5 heads: 252 ÷ 1,024 = 24.61% — about 1 in 4.06.
4. At least 5 heads: (252 + 210 + 120 + 45 + 10 + 1) ÷ 1,024 = 638 ÷ 1,024 = 62.30%.
5. At least one head: 1 − 0.5¹⁰ = 1,023 ÷ 1,024 = 99.90%.

When events aren’t independent

If B depends on A, replace P(B) with the conditional probability P(B | A), the chance of B once A has happened: P(A and B) = P(A) × P(B | A). Drawing two aces from a 52-card deck without putting the first back: 4/52 × 3/51 = 1/221, about 0.45%. With the card put back the draws are independent and the chance is 4/52 × 4/52 = 1/169, about 0.59%.

• Independent is not the same as mutually exclusive. Mutually exclusive events can’t happen together (a single die showing both 2 and 5), so P(A and B) = 0 and P(A or B) = P(A) + P(B). Two events with non-zero chances can’t be both independent and mutually exclusive.
• Past results don’t change independent odds. After five heads in a row the next flip is still 50% heads. The run was unlikely before it started, not the next flip.
• Small chances add up over many tries. A 1% risk per attempt becomes a 63.4% chance of at least one occurrence over 100 independent attempts: 1 − 0.99¹⁰⁰.