Binomial distribution calculator with a bar chart

A binomial distribution counts how many successes you get when the same trial is repeated a set number of times, independently, with the same chance of success every time. If X ~ B(n, p), the chance of exactly k successes is P(X = k) = nCk × pᵏ × (1 − p)ⁿ⁻ᵏ. The chance of exactly 6 heads in 10 fair coin flips is 210 ÷ 1024 = 0.2051. This binomial distribution calculator is part of our Probability Calculator: set n and p, choose exactly, at most, at least or between, and it shades the bars and gives the probability, with the mean and standard deviation.

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How to use the binomial distribution calculator

1. Open the Probability Calculator and choose Binomial at the top. It opens on Normal.
2. Under Distribution, set Trials n and Success chance p.
3. Under Probability, pick a region in Shade and set the bounds, using the table below.
4. Read the probability, written out in full, such as P(X ≤ 3) = 0.1719. The chart shades the bars you have included.
5. Open Formulas for the mean E(X) = np, the variance and the standard deviation with your numbers.
6. To practise, press Random problem in the Practice section.

For a binomial, the calculator includes the bounds themselves, and the result line shows ≤ and ≥ to say so. For fewer than k (X < k), use at most k − 1. For more than k, use at least k + 1.

Binomial distribution formula

nCk counts the orders in which the k successes can happen, pᵏ is the chance of k successes, and (1 − p)ⁿ⁻ᵏ the chance of the n − k failures. The mean and standard deviation are:

When to use a binomial distribution

• A fixed number of trials, n.
• Each trial has two outcomes: success or failure.
• The trials are independent.
• The chance of success, p, is the same every time.

Drawing cards without putting them back fails the last two, because p changes after each draw. Counting until the first success fails the first, because n isn't fixed. For the whole topic, including cumulative probabilities, the normal approximation and hypothesis tests, see the binomial distribution guide.

Worked examples you can check in the calculator

Example 1: exactly 6 heads in 10 flips

Set n = 10 and p = 0.5, choose P(a < X < b), and make a = b = 6: the calculator gives 0.2051.

Example 2: at most 3 heads

P(X ≤ 3) adds the chances of 0, 1, 2 and 3 heads: (1 + 10 + 45 + 120) ÷ 1024 = 176 ÷ 1024 = 0.1719. Choose P(X < a) with a = 3.

Example 3: at least one faulty item

Each item from a machine has a 10% chance of being faulty, independently. In a batch of 15, what is the chance that at least one is faulty? It is easiest through the complement: P(none) = 0.9¹⁵ = 0.2059, so P(at least 1) = 1 − 0.2059 = 0.7941. In the calculator, n = 15, p = 0.1, P(X > a) with a = 1. The mean is 15 × 0.1 = 1.5 faulty items.

Example 4: guessing on a multiple-choice test

A test has 20 questions, each with 4 options, and a student guesses every one, so n = 20 and p = 0.25. The mean is 20 × 0.25 = 5 correct, with standard deviation √(20 × 0.25 × 0.75) = 1.94. The chance of 10 or more correct is P(X ≥ 10) = 0.0139, about 1 in 72: guessing rarely gets you to half marks.

Try a binomial calculation, then press Mark to check your answer in the Probability course.

Quick check

Common mistakes

• Mixing up at least and more than. At least 3 includes 3; more than 3 starts at 4.
• Forgetting nCk. pᵏ(1 − p)ⁿ⁻ᵏ is the chance of one particular order; multiply by the number of orders.
• Adding up dozens of terms for an at least question. Use 1 − P(at most k − 1).
• Using a binomial when p changes, as when sampling without replacement from a small group.
• Rounding each term before adding. Keep full values and round the total.

Read next

Maths · StatisticsBinomial distribution: conditions, formula, mean, varianceWhat the binomial distribution is, when to use it, its formula, mean and variance, cumulative probabilities, the normal approximation and exam questions.