Probability Calculator

Work out the chance of two independent events, repeated trials (binomial probability), or the number of combinations and permutations.

Probability rules

P(not A) = 1 − P(A)
P(A and B) = P(A) × P(B) (independent)
P(A or B) = P(A) + P(B) − P(A and B)
P(neither) = (1 − P(A)) × (1 − P(B))

Binomial probability

P(exactly k) = nCk × p^k × (1 − p)^(n − k)

Example: on a 10-question true/false quiz, the chance of guessing exactly 3 right is 10C3 × 0.5³ × 0.5⁷ = 120 ÷ 1024 ≈ 11.7%. The chance of guessing at least 7 right is about 17.2%.

Combinations and permutations

nCr = n! ÷ (r! × (n − r)!)
nPr = n! ÷ (n − r)!

Choosing 3 people for a committee from 10 gives 10C3 = 120 options. Picking a president, vice president and treasurer from the same 10 gives 10P3 = 720, because the order (who gets which role) now matters.

Probability as odds and percentages

A probability of 0.25 is the same as 25% or "1 in 4". Odds compare successes with failures, so 0.25 is odds of 1 to 3. The calculator shows each result as a decimal, a percentage and a "1 in N" figure.

Last reviewed October 2026.

Frequently asked questions

How do I find the probability of two events both happening?

If the events are independent (one does not affect the other), multiply their probabilities: P(A and B) = P(A) × P(B). The chance of flipping heads twice is 0.5 × 0.5 = 0.25, or 25%.

What is the probability of A or B?

For independent events, P(A or B) = P(A) + P(B) − P(A and B). You subtract the overlap so the case where both happen is not counted twice.

What does "independent" mean?

Two events are independent if knowing one happened does not change the chance of the other. Coin flips and dice rolls are independent. Drawing two cards without putting the first back is not, because the deck has changed.

When should I use the binomial calculator?

Use it when you repeat the same yes/no trial a fixed number of times with the same chance of success each time, such as the number of heads in 10 flips or the number of correct guesses on a multiple-choice quiz.

What is the difference between combinations and permutations?

Combinations (nCr) count selections where order does not matter, like choosing 3 toppings from 10. Permutations (nPr) count arrangements where order matters, like gold, silver and bronze among 10 runners.