Probability Distribution Calculator

Calculate probabilities for binomial, Poisson, normal, and uniform distributions from distribution parameters.

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Distribution

P(X = k)

0.25

The probability of exactly 5 successes in 10 trials with p = 0.5 is 24.6094%.

P(X = k)
0.25
P(X ≤ k)
62.3%
Mean
5
Variance
2.5
Std Dev
1.58

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success.

Also in Probability

Probability & Statistics

Probability distribution calculator: compute PMF, PDF, CDF, and distribution statistics

A probability distribution calculator computes point probabilities, cumulative probabilities, and key statistics for common distributions. It supports binomial, Poisson, normal, and uniform distributions, showing the probability mass or density function value, the CDF, the mean, variance, and standard deviation.

Supported distributions

The binomial distribution models the number of successes in n independent trials, each with probability p. Its PMF gives P(X = k) = C(n,k) × p^k × (1−p)^(n−k).

The Poisson distribution models the number of events in a fixed interval given an average rate λ. Its PMF is P(X = k) = e^(−λ) × λ^k / k!.

The normal (Gaussian) distribution is the continuous bell curve defined by mean μ and standard deviation σ. The calculator computes the PDF value and the CDF using a numerical approximation of the error function.

The continuous uniform distribution assigns equal probability density to all values in the interval [a, b].

Binomial: P(X=k) = C(n,k) × p^k × (1−p)^(n−k)

Probability of exactly k successes in n trials.

Poisson: P(X=k) = e^(−λ) × λ^k / k!

Probability of exactly k events given average rate λ.

Normal CDF: Φ(x) = ½[1 + erf((x−μ) / (σ√2))]

Cumulative probability up to x for a normal distribution.

Frequently asked questions

What is the difference between PMF and PDF?

A probability mass function (PMF) applies to discrete distributions like binomial and Poisson, giving the exact probability of each outcome. A probability density function (PDF) applies to continuous distributions like normal and uniform — it gives density, not probability, for a single point.

Why is the normal PDF value sometimes greater than 1?

The PDF gives probability density, not probability. The probability of any single exact value is zero for a continuous distribution. The PDF can exceed 1 as long as the total area under the curve equals 1.

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