Binomial Distribution Explorer

Drag the number of trials and the success probability and watch the binomial distribution change shape in real time.

Distribution

Number of trials n = 20Success probability p = 0.50

Mean = np = 10; standard deviation = √[np(1−p)] = 2.236.

Look up k = 10

Binomial(n = 20, p = 0.50)

Dark bars (shaded blue): outcomes at most k. The bordered bar is exactly k.

P(X = 10)

0.1762

P(X ≤ 10)

0.5881

P(X ≥ 10)

0.5881

Drag n or p and watch the shape change: small n with p away from 0.5 looks skewed, while larger n, or p near 0.5, looks close to the bell curve the central limit theorem predicts.

How to use this tool

Drag the n slider to set the number of trials.
Drag the p slider to set the probability of success on each trial.
Drag the k slider to shade outcomes up to k and read the three probabilities below the chart.
Compare a small n with p far from 0.5 (skewed) against a larger n or p near 0.5 (closer to bell-shaped).

Formula Reference

Binomial probability

P(X = k) = C(n, k) p^k (1-p)^(n-k)

Mean

Standard deviation

sqrt(np(1-p))

ArticleThe Binomial Distribution ArticleThe Central Limit Theorem, Explained With Worked Examples ArticleThe Normal Distribution CalcBinomial Probability Calculator TableBinomial Table GlossaryGlossary: Binomial Distribution

Frequently Asked Questions

Why does the distribution look skewed for small n and extreme p?+

With few trials and a success probability far from 0.5, one tail of possible outcomes is cut off by the boundary at 0 or n, which pushes the bulk of the probability to one side.

When does the binomial distribution start looking like a normal distribution?+

As n grows, especially when p is not too close to 0 or 1, the shape becomes increasingly symmetric and bell-like. A common rule of thumb is that np and n(1-p) should both be at least 5 or 10 for the normal approximation to work well.

How is this different from the Binomial Probability Calculator?+

The calculator gives one exact numeric answer for chosen values. This tool is for exploring the whole shape interactively and building intuition for how n and p affect it.

What do the shaded bars mean?+

The blue bars are all outcomes from 0 up to the k you have selected, so their total height sums to P(X <= k). The bar with a dark border is the single outcome X = k.

How to use this tool

Drag the n slider to set the number of trials.

Drag the p slider to set the probability of success on each trial.

Drag the k slider to shade outcomes up to k and read the three probabilities below the chart.

Compare a small n with p far from 0.5 (skewed) against a larger n or p near 0.5 (closer to bell-shaped).