Compute exact P(X=k) and cumulative binomial probabilities P(X≤k), P(X≥k), distribution mean, variance, and standard deviation.
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The Binomial Probability Calculator evaluates the probability of achieving exactly or cumulatively k successes in n independent Bernoulli trials, where each trial has the same probability of success p.
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, \quad \mu = np, \quad \sigma^2 = np(1-p)Where n is number of trials, k is number of successes, p is success probability per trial, and (1-p) is failure probability.
Scenario: A basketball player with an 80% free throw percentage (p = 0.8) takes 10 shots. Find the probability of making exactly 8 shots.
Identify parameters: n = 10, p = 0.8, k = 8.
Compute combinations: C(10, 8) = 45.
Calculate powers: 0.8^8 ≈ 0.16777, and (1 - 0.8)^2 = 0.2^2 = 0.04.
Multiply: 45 × 0.16777 × 0.04 = 0.30199 (30.20%).
A Bernoulli trial is a random experiment with exactly two mutually exclusive outcomes, conventionally labeled "success" and "failure", where the probability of success remains constant.
To find P(X ≥ k), sum the probabilities of obtaining k, k+1, ..., up to n successes, or calculate 1 - P(X ≤ k - 1).
Use the Poisson distribution when events occur continuously over time or space with a known average rate λ, or as an approximation for binomial experiments where n is very large (n > 100) and p is very small (p < 0.01).
The mean (expected value) is μ = n × p. For example, rolling a die 60 times with p = 1/6 for a six gives an expected value of 60 × (1/6) = 10 sixes.
Calculate the probability of single events, union of events, intersection, conditional probability P(A|B), and complements.
Calculate probabilities for coin toss trials, exact heads/tails count, at least k heads, and consecutive streak probabilities.
Calculate Poisson probability P(X=k), cumulative event odds, and arrival rates per time or spatial interval.