Probability Calculator
Calculate combinations (nCr), permutations (nPr), basic event probability, and binomial distribution — exact and cumulative probabilities in your browser.
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Combinations C(n, r)
Number of ways to choose r items from n items where order does not matter.
C(n,r) = n! / (r! × (n−r)!)
Permutations P(n, r)
Number of ways to arrange r items from n items where order matters.
P(n,r) = n! / (n−r)!
Basic Event Probability
Probability = favorable outcomes ÷ total outcomes.
Binomial Distribution
P(X=k) = C(n,k) × pk × (1−p)n−k
How It Works
Combinations vs Permutations
Combinations C(n,r) — choosing r items from n where order does not matter. Example: choosing 3 students from a class of 10 gives C(10,3) = 120 ways. Permutations P(n,r) — arranging r items from n where order matters. Example: awarding gold/silver/bronze from 10 athletes gives P(10,3) = 720 ways.
Binomial Distribution
Models the number of successes k in n independent trials each with probability p. The exact probability P(X=k) uses the binomial coefficient C(n,k) multiplied by p^k × (1−p)^(n−k). The cumulative P(X≤k) sums all exact probabilities from 0 to k. Mean = n·p, Standard deviation = √(n·p·(1−p)).
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Open calculator →FAQ
What is the difference between combinations and permutations?
Combinations C(n,r) count the number of ways to choose r items from n items where order does NOT matter (e.g., selecting a committee). Permutations P(n,r) count the number of ways to arrange r items from n where order DOES matter (e.g., first, second, and third place). C(n,r) = n! / (r!(n−r)!), P(n,r) = n! / (n−r)!.
What is a binomial distribution?
The binomial distribution models the number of successes in n independent trials, each with probability p of success. P(X=k) = C(n,k) × p^k × (1−p)^(n−k). The mean is n×p and the standard deviation is √(n×p×(1−p)). Used in quality control, polling, and clinical trials.
How do you calculate the probability of an event?
Probability = favorable outcomes ÷ total equally likely outcomes. For example, the probability of drawing an ace from a standard deck is 4/52 ≈ 7.69%. The complement (event NOT occurring) is 1 − P(event). Odds represent the ratio of favorable to unfavorable outcomes.