Binomial distribution calculator
Enter your values. The answer and the full worked solution update as you type.
Result
| k | P(X = k) | P(X ≤ k) |
|---|---|---|
| 0 | 0.02825 | 0.02825 |
| 1 | 0.1211 | 0.1493 |
| 2 | 0.2335 | 0.3828 |
| 3 | 0.2668 | 0.6496 |
| 4 | 0.2001 | 0.8497 |
| 5 | 0.1029 | 0.9527 |
| 6 | 0.03676 | 0.9894 |
| 7 | 0.009002 | 0.9984 |
| 8 | 0.001447 | 0.9999 |
| 9 | 0.0001378 | 1 |
| 10 | 5.905 × 10⁻⁶ | 1 |
Work out binomial probabilities for n independent trials that each succeed with probability p: exactly x successes, at most x, at least x, fewer than x or more than x. The calculator shows the binomial formula, every term it adds (or the shorter complement), the mean np and the standard deviation √(np(1 − p)). For n up to 30 it also lists the whole distribution in a table.
Step-by-step solution
Binomial probability formula
Add the probabilities
Mean and standard deviation
Key formulas
- Binomial
- P(X = k) = C(n,k)·pk(1 − p)n−k
- Complement
- P(not A) = 1 − P(A)
How to calculate binomial probabilities
- Check that the situation is binomial: a fixed number of trials n, two outcomes per trial, the same probability of success p every time and independent trials.
- For exactly x successes use P(X = x) = C(n, x)·pˣ·(1 − p)ⁿ⁻ˣ, where C(n, x) = n!/(x!(n − x)!) counts the orders in which the successes can occur.
- For “at most x”, add P(X = 0) up to P(X = x). For “at least x”, add P(X = x) up to P(X = n).
- If the sum has many terms, use the complement: P(X ≥ x) = 1 − P(X ≤ x − 1).
- Translate the words carefully: “fewer than 4” is X ≤ 3 and “more than 4” is X ≥ 5.
Questions and answers
What are the mean and standard deviation of a binomial distribution?
The mean is μ = np and the variance is σ² = np(1 − p), so the standard deviation is σ = √(np(1 − p)). For n = 10 and p = 0.3 that gives μ = 3 and σ = 1.449.
What is the difference between “at least” and “more than”?
“At least 3” includes 3: P(X ≥ 3). “More than 3” does not: P(X > 3) = P(X ≥ 4). For a discrete distribution these are different numbers, so choose the option that matches the wording of the question.
Why does the calculator sometimes use the complement?
Because all the probabilities add up to 1, P(X ≥ x) = 1 − P(X ≤ x − 1). When the complement has fewer terms it is quicker by hand, and the steps show that route. The result is the same.
How large can n be?
Up to 10 000 trials. The terms are computed with logarithms, so the large factorials do not overflow and the probabilities stay accurate.
When is the binomial distribution not the right model?
When the trials are not independent or p changes, for example when you draw without replacement from a small group: that follows the hypergeometric distribution. When n is large and p is small, the Poisson distribution with λ = np is a good approximation.
Practise it yourself
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