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.

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

Name: Binomial Distribution Calculator
Availability: InStock
Author: Nham Vu

Enter number of trials, success probability, and successes to compute exact and cumulative binomial probabilities.

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

Results

P(X = k) —

P(X ≤ k) (lower tail) —

P(X ≥ k) (upper tail) —

Mean μ = np —

Std Dev σ = √(np(1-p)) —

Probability Mass Function

Enter parameters and click Calculate

PMF Table

Results will appear here

Summary