Mathematics, Statistics & Geometry

Binomial Distribution Calculator

Calculate the probability of exactly k successes in n trials. Find the mean, variance, and standard deviation of a binomial experiment.

P(X = 5)
24.609
Expected Value (Mean)5
Variance2.5
Std. Deviation1.581
Calculation StepsFormula: P(X=k) = (n! / (k!(n-k)!)) * p^k * (1-p)^(n-k) n = 10, k = 5, p = 0.5 Combinations C(10,5) = 252 Probability = 252 * 0.5^5 * 0.5000^5 = 0.24609375

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Calculate Success and Failure

The Binomial Distribution calculator is a fundamental tool for probability and statistics. Whether you are flipping a coin, testing manufactured parts for defects, or predicting the outcome of a binary event, this tool provides the exact probability of achieving your target results.

P(X=k)=(nk)pk(1−p)n−k\begin{aligned} P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \end{aligned}

Where:
n=
The total number of independent experiments
p=
The probability of success on a single trial
k=
The specific number of successes you want the probability for
(nk)\binom{n}{k}=
The number of ways to choose k successes from n trials

When to Use This Calculator

Use the binomial distribution when your problem meets these four criteria:

  1. The number of trials (n) is fixed in advance.
  2. Each trial has only two possible outcomes (e.g., Yes/No, Success/Failure, Heads/Tails).
  3. The trials are independent (the result of one doesn't affect the next).
  4. The probability of success (p) remains constant for every trial.

Statistics Provided

  • P(X=k): The probability of getting exactly k successes.
  • Mean (Expected Value): The average number of successes you would expect over many repetitions.
  • Variance & Std. Deviation: Measures of how much the results are likely to spread out from the mean.

Real-World Use Cases

  • Quality Control: Finding the probability that a batch of 100 items contains exactly 3 defects.
  • Marketing: Estimating the number of sales based on a known conversion rate and a fixed number of website visitors.
  • Sports: Calculating the probability of a basketball player making 8 out of 10 free throws based on their career percentage.
  • Genetics: Predicting the probability of offspring inheriting a specific trait.

Frequently Asked Questions

A binomial distribution describes the number of successes in a fixed number of independent trials, where each trial has the same probability of success and only two possible outcomes (success or failure).

Multiply the binomial coefficient (n choose k) by the success probability raised to the power of k, and the failure probability raised to the power of n-k.

  1. Fixed number of trials. 2) Each trial is independent. 3) Only two outcomes. 4) Constant probability of success.

The mean (expected value) is simply n × p.

Binomial is discrete (counting whole successes), while normal is continuous. However, for large n, the binomial distribution can be approximated by a normal distribution.

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