What is the binomial distribution?

The binomial distribution describes the number of successes k in n independent Bernoulli trials, each with success probability p. It applies when trials are independent, there are exactly two outcomes per trial (success/failure), and p is constant across all trials.

What is the formula used?

P(X=k) = C(n,k) × p^k × (1-p)^(n-k), where C(n,k) = n! / (k! × (n-k)!) is the binomial coefficient. The cumulative P(X≤k) sums P(X=i) for i from 0 to k.

What is the difference between P(X=k), P(X≤k), and P(X≥k)?

P(X=k) is the exact probability of getting exactly k successes. P(X≤k) is the cumulative probability of getting k or fewer successes. P(X≥k) is the probability of getting k or more successes, equal to 1 − P(X≤k−1).

What are the mean and standard deviation of the binomial distribution?

The mean (expected value) is μ = n × p. The standard deviation is σ = √(n × p × (1 − p)). These are displayed alongside the probability results.

How do I calculate "at least", "at most" and "between" probabilities?

At most k: P(X ≤ k) = P(0) + … + P(k). Fewer than k: P(X k) = 1 − P(X ≤ k). Between a and b inclusive: P(a ≤ X ≤ b) = P(X ≤ b) − P(X ≤ a − 1). With n = 10 and p = 0.5, P(3 ≤ X ≤ 7) = 0.890625.

When can I use the normal approximation to the binomial?

A common rule is np ≥ 10 and n(1 − p) ≥ 10 (some textbooks use 5). Then X is approximately normal with mean np and standard deviation √(np(1 − p)). Use a continuity correction: P(X ≤ k) ≈ Φ((k + 0.5 − np) ÷ σ). For n = 100 and p = 0.5, P(X ≤ 55) is 0.8644 exact and 0.8643 by the approximation.

This calculator supports n up to 1000. For very large n, log-space computation is used internally to avoid floating-point overflow when computing factorials.

Is this tool accurate for small p or large n?

Yes. The tool uses log-gamma / log-sum-exp arithmetic to maintain precision even when p is very small (e.g., 0.001) or n is large (e.g., 500).

Binomial Distribution Calculator

Enter n, p and k (plus an optional range a to b) to get exact, at most, less than, at least, more than and between probabilities with the worked formula.

P(X = k) = C(n, k) × p^k × (1 − p)^(n − k), where C(n, k) = n! ÷ (k!(n − k)!).

Parameters

Number of trials (n)

Integer from 1 to 1000

Probability of success (p)

Decimal between 0 and 1 (e.g., 0.3 for 30%)

Number of successes (k)

Integer from 0 to n

Range for P(a ≤ X ≤ b) (optional)

Inclusive integers; leave blank to skip

Results

P(X = k) —

P(X ≤ k) (lower tail) —

P(X ≥ k) (upper tail) —

P(X < k) (fewer than) —

P(X > k) (more than) —

P(a ≤ X ≤ b) —

Mean μ = np —

Variance σ² = np(1-p) —

Std Dev σ = √(np(1-p)) —

Worked Formula

Normal Approximation Check

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Probability Mass Function

Enter parameters and click Calculate

PMF Table

Results will appear here

Quick answer

P(X = k) = C(n, k) × p^k × (1 − p)^(n − k), where C(n, k) = n! ÷ (k!(n − k)!). Add these terms for cumulative answers: "at most k" is P(X ≤ k), "fewer than k" is P(X ≤ k − 1), "at least k" is 1 − P(X ≤ k − 1), and "between a and b" is P(X ≤ b) − P(X ≤ a − 1). The mean is np and the variance np(1 − p).

Examples

n = 10, p = 0.5, k = 5	Probability
P(X = 5)	0.246094
P(X ≤ 5) at most	0.623047
P(X < 5) fewer than	0.376953
P(X ≥ 5) at least	0.623047
P(3 ≤ X ≤ 7) between	0.890625

Enter n, p, k and an optional range to get all six probabilities, mean, variance, the chart and the full table.

Summary

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. Enter the number of trials (n), the success probability (p) and the number of successes (k). The calculator returns the exact probability P(X=k) and every cumulative form: at most, fewer than, at least and more than. An optional range gives P(a ≤ X ≤ b). It also shows the mean, variance and standard deviation, the formula with your numbers, a normal-approximation check, a probability bar chart and the full PMF and CDF table.

How it works

Enter the number of trials (n) — a positive integer up to 1000.
Enter the probability of success per trial (p) — a value between 0 and 1.
Enter the number of successes (k) you want to evaluate — an integer from 0 to n.
Optionally enter a range a to b to get the probability that X falls between them, inclusive.
The calculator computes P(X=k) = C(n,k) × p^k × (1−p)^(n−k) and sums these terms for the cumulative and between probabilities.
The worked formula, mean, variance, normal-approximation check, chart and full table update instantly.

Use cases

Quality control: find the probability of at most 2 defective items in a batch of 50.
Clinical trials: estimate the chance of observing k or more responders out of n patients.
A/B testing: calculate the likelihood of achieving a conversion rate result by chance.
Genetics: compute inheritance probabilities across offspring.
Exam prep: solve textbook binomial distribution problems with step-by-step formulas.
Finance: model binary outcome scenarios (up/down moves in discrete-time models).
Sports analytics: estimate win probabilities over a series of independent games.
Customer surveys: find the probability of exactly k positive responses from n respondents.

Frequently Asked Questions

Last updated: 2026-10-10 · Reviewed by Nham Vu