Enter the number of trials (n), probability of success (p), and number of successes (x) to compute binomial probabilities. Your results include P(X = x), cumulative probabilities P(X ≤ x), P(X ≥ x), P(X < x), and P(X > x), plus the mean and standard deviation of the distribution.
P(X = x)
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P(X < x)
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P(X ≤ x)
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P(X > x)
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P(X ≥ x)
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Mean (μ = np)
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Standard Deviation (σ)
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A binomial experiment is a statistical experiment that has exactly two possible outcomes on each trial — typically called 'success' and 'failure'. It must consist of a fixed number of independent trials, each with the same probability of success. Examples include flipping a coin a set number of times or testing whether manufactured items are defective.
The binomial distribution describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success p. It is characterized by two parameters: n (number of trials) and p (probability of success). The distribution is denoted X ~ Bin(n, p).
The probability of exactly x successes in n trials is given by the formula P(X = x) = C(n, x) × p^x × (1−p)^(n−x), where C(n, x) is the binomial coefficient (n choose x). This calculator handles all the computation automatically — just enter n, p, and x.
The number of trials n is the total count of independent experiments performed. For example, if you flip a coin 20 times, n = 20. Each trial must be independent, and the probability of success must remain constant across all trials.
The probability of success p is the likelihood that any single trial results in a success. It must be a value between 0 and 1. For a fair coin flip, p = 0.5. For a biased die showing a six, p ≈ 0.167.
Cumulative binomial probability refers to the probability that the number of successes falls within a certain range. For example, P(X ≤ x) is the probability of getting at most x successes, calculated by summing P(X = k) for all k from 0 to x. This calculator outputs P(X < x), P(X ≤ x), P(X > x), and P(X ≥ x).
For a binomial distribution with parameters n and p, the mean is μ = n × p and the standard deviation is σ = √(n × p × (1 − p)). These describe the center and spread of the distribution. For example, with n = 20 and p = 0.5, the mean is 10 and σ ≈ 2.24.
Use the binomial distribution when you have a fixed number of independent trials, each with exactly two outcomes and a constant probability of success. If trials are not independent or the probability changes, consider the hypergeometric or other distributions. For very large n with small p, the Poisson distribution is a useful approximation.