Probability Distribution Calculator

Binomial, normal and Poisson, with cumulative values.

Distribution

First parameter

Binomial: n · Normal: mean · Poisson: lambda

Second parameter

Binomial: p · Normal: sd · Poisson: k

Third parameter

Binomial: k · Normal: x · Poisson: leave blank

Press Calculate.

Choosing the distribution

Use	When	Example
Binomial	A fixed number of independent trials, each a success or failure with constant probability	Heads in 10 coin tosses
Normal	A continuous measurement clustering symmetrically about a mean	Heights, measurement error, exam scores
Poisson	Counts of events in a fixed interval, at a constant average rate, occurring independently	Calls per hour, typos per page

Binomial

P(X = k) = C(n, k) × p^k × (1 − p)^{n − k}

C(n, k) counts the arrangements — how many different ways k successes can fall among n trials. Mean is np, variance np(1 − p).

Normal

The z score, (x − mean) ÷ sd, restates any normal value in standard deviations from the mean. That is what makes one table serve every normal distribution. The cumulative probability here uses an error-function approximation accurate to about seven decimal places — ample for coursework, and worth knowing if you are comparing against printed tables that round to four.

Poisson

P(X = k) = e^−λ × λ^k ÷ k!

Its distinguishing property: mean and variance are both λ. If your data has a variance much larger than its mean, it is overdispersed and Poisson is the wrong model.

Exact versus cumulative

P(X = k) is the probability of exactly k. P(X ≤ k) sums every outcome up to and including k. Questions asking 'at most', 'at least' or 'fewer than' want a cumulative value, and 'at least k' is 1 − P(X ≤ k − 1) — note the k − 1. Off-by-one here is the most common error in this topic.