Number of successes in n independent Bernoulli trials at probability p.
A free, animated binomial distribution you can read here or embed on any website, from Scrollchart.
Binomial Distribution
The Binomial distribution gives the probability of observing exactly k successes in n independent Bernoulli trials, each with probability p of success. The diagram displays this as a discrete bar histogram: each bar sits above an integer k value and its height equals P(X = k) = C(n, k) * p^k * (1-p)^(n-k).\n\nWith n = 20 and p = 0.4, the mean is n * p = 8 and the most probable outcome is 8 successes. The distribution is symmetric when p = 0.5 and skewed right when p < 0.5, skewed left when p > 0.5. As n grows large, the histogram converges toward the shape of a normal distribution with mean np and standard deviation sqrt(np(1-p)), which is why a dashed normal-approximation overlay is optionally shown.
Probability mass function: P(X = k) = C(n, k) * p^k * (1-p)^(n-k), where C(n, k) = n! / (k! * (n-k)!) is the binomial coefficient counting the number of ways to choose k successes from n trials.\n\nMean: n * p. Variance: n * p * (1 - p). Standard deviation: sqrt(n * p * (1 - p)). For n = 20, p = 0.4: mean = 8, standard deviation = sqrt(20 * 0.4 * 0.6) = sqrt(4.8) = approximately 2.19.\n\nNormal approximation: for large n the Binomial is well approximated by Normal(mean = np, variance = np(1-p)). The rule of thumb requires both np >= 5 and n(1-p) >= 5. Continuity correction (shifting by 0.5) improves accuracy for moderate n.
Ars Conjectandi , Bernoulli, J. (1713) - Posthumous publication that introduced the law of large numbers and laid the foundations for the binomial distribution.
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Source & accuracy
This binomial distribution is an editorial illustration built to represent the concept accurately. Where it shows figures, they are typical or representative values chosen to make the relationship clear, not a single underlying dataset. The diagram and its explainer are reviewed and maintained centrally, and updated over time as understanding improves.
Counting successes across independent trials
The binomial distribution models the count of successes in n independent Bernoulli trials, each with the same probability p of success. For example, flipping a fair coin 10 times and counting heads follows a binomial distribution with n = 10 and p = 0.5.
The distribution answers questions like: if we make 100 sales calls with a 5 percent conversion rate, what is the probability of closing at least 3 deals? The histogram shows all possible outcomes (0 through n successes) and their frequencies.
Shape variation with probability and sample size
When p = 0.5, the binomial distribution is symmetric and bell-shaped, especially for moderate to large n. When p deviates from 0.5 (toward 0 or 1), the distribution skews; smaller n values produce choppier, more discrete histograms, while larger n creates smoother, more normal-like shapes.
The mean is n times p, and the variance is n times p times (1 - p). As n grows, the binomial distribution increasingly resembles a normal distribution, a phenomenon known as the central limit theorem.
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Reference
What this is
A free, embeddable, animated binomial distribution for any website.
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