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Calculator Answers

Binomial Distribution Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A binomial distribution calculator is a probability tool that computes discrete probabilities for experiments with independent, binary outcomes. Given a number of trials (n), a success rate per trial (p), and target successes (k), it calculates the exact probability P(X = k) using the binomial formula. It also sums individual probabilities to yield cumulative results like P(X ≤ k) and standard statistics like mean and variance.

Compute individual and cumulative binomial probabilities for any number of trials and success rates.

See also: Binomial Probability Calculator and Poisson Distribution Calculator.

Quick Answer

Solve discrete binomial probability distributions. Enter trials, success probability, and target successes to see PMF and cumulative results.

Binomial Parameters

Total number of independent events (0 to 170). · e.g. 10

Success rate per trial (0 to 1). · e.g. 0.5

The target successes to solve (0 to n). · e.g. 5

Distribution Results

Probability of exactly 5 successes: P(X = 5)

0.246094

At most 5 successes: P(X ≤ 5)0.623047
Fewer than 5 successes: P(X < 5)0.376953
At least 5 successes: P(X ≥ 5)0.623047
More than 5 successes: P(X > 5)0.376953
Detailed Steps and Stats1. Calculate combination value nCr: C(10, 5) = 10! / (5! × (10 − 5)!) = 252 2. Calculate probability P(X = 5): P(X = 5) = C(10, 5) × p^5 × (1 − p)^(10 − 5) = 252 × 0.5^5 × 0.5^5 = 0.246094 3. Distribution Parameters: Mean (μ) = n × p = 5 Variance (σ²) = n × p × (1−p) = 2.5 Standard Deviation (σ) = √Variance = 1.5811

Binomial distribution outputs are discrete. In contrast, the normal distribution is a continuous approximation of the binomial distribution.

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Examples

10 trials, p = 0.5, success k = 5

P(X = 5) ≈ 0.2461, P(X ≤ 5) ≈ 0.6230

5 trials, p = 0.2, success k = 1

P(X = 1) ≈ 0.4096, P(X ≤ 1) ≈ 0.7373

How it works

The calculator determines the combinations of selecting k successes from n trials, and multiplies it by the success and failure probability rates.

PMF Formula · P(X = k) = nCr × p^k × (1−p)^(n−k)

How to calculate binomial probability

To solve binomial distribution problems, determine the parameters:

  • n: The number of trials.
  • p: The probability of success on a single trial.
  • k: The number of successes you want to evaluate.

Plug these into the probability mass function:

P(X = k) = (n! / (k! × (n − k)!)) × p^k × (1 − p)^(n − k)

Related probability and statistics calculators

Solve other statistical distributions using these tools:

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Binomial Probability CalculatorCalculate binomial probability for n trials, k successes, and per-trial probability p. Supports exact, at-most, and at-least modes.Poisson Distribution CalculatorCalculate Poisson probabilities for an average rate lambda and event count k. Supports exact, at-most, and at-least modes.

Frequently asked questions

A binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials (n), where each trial has the same probability of success (p).

The probability of getting exactly k successes in n trials is: P(X = k) = C(n, k) × p^k × (1 − p)^(n − k). Here, C(n, k) is the combination formula n! ÷ (k!(n − k)!).

Individual probability P(X = k) finds the chance of getting exactly k successes. Cumulative probability P(X ≤ k) finds the chance of getting k or fewer successes, which is the sum of probabilities for 0, 1, ..., k successes.

A binomial experiment must satisfy four conditions: 1) Fixed number of trials (n). 2) Each trial has only two outcomes (success/failure). 3) Trials are independent. 4) The probability of success (p) is constant.

The mean (expected value) is μ = n × p. The variance is σ² = n × p × (1 − p). The standard deviation is the square root of the variance: σ = √(n × p × (1 − p)).