Enter the number of trials, probability of success, and number of successes (x) to compute binomial probabilities. The Binomial Distribution Calculator returns P(X = x), P(X ≤ x), P(X ≥ x), P(X < x), and P(X > x), along with the mean and standard deviation of the distribution. A probability mass table is also shown for every possible outcome.
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 a fixed number of independent trials, each with exactly two possible outcomes — success or failure. The probability of success remains constant across every trial. Examples include flipping a coin 10 times or testing 50 products for defects.
The binomial distribution describes the number of successes in a fixed number of independent Bernoulli trials. It is defined by two parameters: n (number of trials) and p (probability of success per trial). The distribution gives the probability of obtaining exactly k successes out of n trials.
Binomial probability is calculated using 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), p is the probability of success, and (1−p) is the probability of failure. The calculator handles this automatically once you enter n, p, and x.
P(X = x) is the exact probability of getting precisely x successes. Cumulative probabilities like P(X ≤ x) add up all probabilities from 0 to x, while P(X ≥ x) sums from x to n. Cumulative probabilities are useful when you want to know the likelihood of getting at most or at least a certain number of successes.
The mean (expected value) of a binomial distribution is μ = n × p. For example, if you flip a fair coin (p = 0.5) 20 times, the expected number of heads is 20 × 0.5 = 10. It represents the average outcome you would expect over many repeated experiments.
The standard deviation is σ = √(n × p × (1 − p)). It measures the spread of the distribution around the mean. A higher standard deviation means more variability in the number of successes across trials.
The number of trials n is the total count of independent repetitions in your binomial experiment. For instance, if you test whether 30 randomly selected voters support a candidate, n = 30. The value of n must be a positive integer.
The probability of success p must be a number between 0 and 1 (inclusive). A p of 0 means success is impossible, while p = 1 means success is certain. Most real-world scenarios use a value somewhere in between, such as 0.3 for a 30% chance of success per trial.