# Binomial Distribution Calculator

Free binomial distribution calculator. Calculate exact, cumulative, and complementary binomial probabilities with P(X=k), P(X≤k), and P(X≥k).

## What this calculates

Calculate binomial probabilities for a fixed number of independent trials. Find the probability of exactly k successes, at most k successes, or at least k successes.

## Inputs

- **Number of Trials (n)** — min 1, max 170 — The total number of independent trials.
- **Number of Successes (k)** — min 0 — The number of successes you want to find the probability for.
- **Probability of Success (p)** — min 0, max 1 — The probability of success on each trial (between 0 and 1).

## Outputs

- **P(X = k)** — Probability of exactly k successes.
- **P(X ≤ k)** — Probability of k or fewer successes.
- **P(X ≥ k)** — Probability of k or more successes.
- **Expected Value E(X)** — The expected number of successes (n × p).
- **Standard Deviation** — Standard deviation of the distribution √(n × p × (1-p)).
- **Formula** — formatted as text — The binomial probability formula used.

## Details

The binomial distribution models the number of successes in n independent trials, each with probability p.

PMF (Probability Mass Function):
P(X = k) = C(n,k) × p^k × (1-p)^(n-k)

Where C(n,k) = n! / (k!(n-k)!) is the binomial coefficient.

Key Properties

- Mean: E(X) = n × p

- Variance: Var(X) = n × p × (1-p)

- Standard deviation: σ = √(n × p × (1-p))

Examples

- Coin flips (p = 0.5)

- Quality control defects

- Survey yes/no responses

## Frequently Asked Questions

**Q: What are the conditions for a binomial distribution?**

A: A binomial distribution requires: (1) a fixed number of trials n, (2) each trial has exactly two outcomes (success/failure), (3) the probability of success p is constant for each trial, and (4) trials are independent of each other.

**Q: What is the difference between P(X = k) and P(X ≤ k)?**

A: P(X = k) is the probability of exactly k successes. P(X ≤ k) is the cumulative probability of k or fewer successes, which is the sum of P(X = 0) + P(X = 1) + ... + P(X = k). Use the cumulative form when you want 'at most' k successes.

**Q: When does the binomial distribution approximate the normal distribution?**

A: When both n × p ≥ 5 and n × (1-p) ≥ 5, the binomial distribution can be well approximated by a normal distribution with mean n × p and standard deviation √(n × p × (1-p)). This is useful for very large n where exact calculations are computationally expensive.

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Source: https://vastcalc.com/calculators/statistics/binomial-distribution
Category: Statistics
Last updated: 2026-04-21