Math & statistics
Binomial Distribution Calculator
Exactly, at most, at least, or between — every binomial probability with the distribution drawn and the calculator syntax (binompdf / binomcdf) shown for your homework.
P(X = 6)
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mean μ = np
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SD σ = √(np(1−p))
Frequently asked questions
What is the difference between binompdf and binomcdf?
binompdf(n, p, k) gives P(X = k), the probability of exactly k successes. binomcdf(n, p, k) gives P(X ≤ k), the cumulative probability of k or fewer. “At least k” is 1 − binomcdf(n, p, k−1).
When can I use the binomial distribution?
When the four BINS conditions hold: Binary outcomes (success/failure), Independent trials, a fixed Number of trials n, and the Same probability p on every trial. Sampling without replacement is approximately binomial when the sample is under 10% of the population.
How do I find “at least one” probabilities?
Use the complement: P(at least 1) = 1 − P(none) = 1 − (1−p)ⁿ. For example the chance of at least one six in 4 dice rolls is 1 − (5/6)⁴ ≈ 0.518.
What are the mean and standard deviation of a binomial distribution?
Mean μ = np and standard deviation σ = √(np(1−p)). For n = 20, p = 0.3: μ = 6 and σ ≈ 2.05 — both shown automatically above.
When is a binomial distribution approximately normal?
The usual Large Counts condition: np ≥ 10 and n(1−p) ≥ 10. Then X is approximately Normal(np, √(np(1−p))), which is how the normal approximation questions on AP Statistics work.
Method: exact binomial probabilities P(X = k) = C(n,k)·pᵏ·(1−p)ⁿ⁻ᵏ computed in log space for numerical stability — the same values binompdf/binomcdf return on a graphing calculator.