The standardised value z = (x − μ) / σ, telling you how many standard deviations x sits from the mean. z = 1.96 is the two-tailed 95% critical value.

What is the difference between P(X ≤ x) and P(X ≥ x)?

They are the left and right tails and add up to 1. For a continuous distribution the probability of an exact value is 0, so including the endpoint makes no difference.

When should I use binomial rather than Poisson?

Use the binomial distribution for a fixed number of independent trials (ten coin flips); use Poisson for counts of events per unit of time or space (customers per hour). They agree closely when n is large and p small.

What are the mean and variance of a binomial distribution?

Mean np and variance np(1−p); both are shown, and the square root of the variance is the standard deviation.

Can I see the distribution as a graph?

Yes. Under the results, click “Plot the normal curve” or “Plot the distribution”: the normal case draws the density curve for your μ and σ, while binomial and Poisson plot the probability of every k as points you can zoom and read off.

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🎰Probability Calculator - Normal, Binomial and Poisson Distributions

The three distributions you need most, in one place: the normal distribution with its z-score, P(X≤x), P(X≥x), two-tailed probability and common percentiles; the binomial distribution with exactly, at most and at least k successes; and the Poisson distribution, each with its mean and variance.

Free · No sign-up · Runs in your browserLast updated: 2026-10-26

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How to use Probability Distributions

1Choose the normal, binomial or Poisson distribution
2Enter the parameters: μ and σ, or n and p, or λ
3Enter the value x or the count k to read the probabilities

FAQ

What is a z-score?: The standardised value z = (x − μ) / σ, telling you how many standard deviations x sits from the mean. z = 1.96 is the two-tailed 95% critical value.
What is the difference between P(X ≤ x) and P(X ≥ x)?: They are the left and right tails and add up to 1. For a continuous distribution the probability of an exact value is 0, so including the endpoint makes no difference.
When should I use binomial rather than Poisson?: Use the binomial distribution for a fixed number of independent trials (ten coin flips); use Poisson for counts of events per unit of time or space (customers per hour). They agree closely when n is large and p small.
What are the mean and variance of a binomial distribution?: Mean np and variance np(1−p); both are shown, and the square root of the variance is the standard deviation.
Can I see the distribution as a graph?: Yes. Under the results, click “Plot the normal curve” or “Plot the distribution”: the normal case draws the density curve for your μ and σ, while binomial and Poisson plot the probability of every k as points you can zoom and read off.