# Probability Calculator - Normal, Binomial and Poisson Distributions

> Free online probability calculator: z-scores and cumulative normal probabilities, P(X=k) and P(X≤k) for binomial and Poisson, plus the mean and variance.

- URL: https://qilingtools.com/en/math/probability
- Category: Math Tools (https://qilingtools.com/en/math)
- Tags: [Math](https://qilingtools.com/en/tags/math), [Statistics](https://qilingtools.com/en/tags/statistics)
- Last updated: 2026-10-26
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

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.

## How to use

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

## Features

- Normal probabilities computed exactly with the error function — no tables needed
- z-score, one- and two-tailed probabilities and P90/P95/P99 percentiles
- Binomial P(X=k), P(X≤k), P(X≥k) with mean and variance
- Poisson computed in log space so large λ or k never overflow
- Probabilities also shown as percentages, ready for a report

## 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.

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