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Binomial Distribution Calculator

Calculate the probability of getting exactly k successes in n independent trials, each with success probability p.

Reviewed by Chase FloiedUpdated

This free online binomial distribution calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Total number of independent trials.

Desired number of successes.

Probability of success on each trial (0 to 1).

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Binomial Distribution Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Binomial Distribution Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

Formula Reference

Binomial Distribution Calculator Formula

See calculator inputs for the governing equation

Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.

When to Use This Calculator

  • •Use the Binomial Distribution Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

About This Calculator

The Binomial Distribution Calculator is a free mathematical calculation tool for students, educators, and professionals who need quick, reliable results. Calculate the probability of getting exactly k successes in n independent trials, each with success probability p. The underlying algorithms implement well-established mathematical formulas and numerical methods. Results are computed instantly in the browser. This tool is useful for learning, verification of hand calculations, and rapid exploration of mathematical relationships. All computation happens locally — no data is sent to a server.

About Binomial Distribution Calculator

The binomial distribution calculator computes the probability of obtaining exactly k successes in n independent trials, where each trial has the same probability of success p. This distribution models coin flips, pass/fail inspections, survey responses (yes/no), medical treatment outcomes, and many other binary processes. It is one of the most widely used discrete probability distributions. The calculator also provides the mean (np) and standard deviation (sqrt(np(1-p))) of the distribution, which help characterize its center and spread. When n is large and p is not too close to 0 or 1, the binomial distribution is well approximated by the normal distribution.

The Math Behind It

The binomial distribution arises from n independent Bernoulli trials, each with success probability p. The probability mass function is P(X=k) = C(n,k)*p^k*(1-p)^(n-k), where C(n,k) is the binomial coefficient. The mean is E(X) = np, the variance is Var(X) = np(1-p), and the standard deviation is sqrt(np(1-p)). The distribution is symmetric when p = 0.5 and skewed otherwise. As n increases with p fixed, the distribution approaches a normal distribution by the central limit theorem (the rule of thumb is np >= 5 and n(1-p) >= 5). When p is very small and n is very large, the binomial converges to the Poisson distribution with lambda = np. The moment generating function is M(t) = (1-p+p*e^t)^n. Cumulative probabilities P(X <= k) are computed by summing the PMF from 0 to k and are available in statistical tables and software. The binomial distribution is the foundation for binomial tests, confidence intervals for proportions, and process capability analysis in quality control.

Formula Reference

Binomial Probability

P(X=k) = C(n,k) * p^k * (1-p)^(n-k)

Variables: n = trials; k = successes; p = probability of success; C(n,k) = n!/(k!(n-k)!)

Worked Examples

Example 1: Quality control inspection

A batch has a 5% defect rate. In a sample of 20 items, what is the probability of exactly 2 defects?

Step 1:n = 20, k = 2, p = 0.05.
Step 2:C(20,2) = 190.
Step 3:P(X=2) = 190 * 0.05^2 * 0.95^18 = 190 * 0.0025 * 0.3972 = 0.1887.

There is about an 18.9% chance of finding exactly 2 defective items.

Example 2: Free throw shooting

A basketball player makes 80% of free throws. What is the probability of making exactly 8 out of 10?

Step 1:n = 10, k = 8, p = 0.8.
Step 2:C(10,8) = 45.
Step 3:P(X=8) = 45 * 0.8^8 * 0.2^2 = 45 * 0.16777 * 0.04 = 0.3020.

The probability of exactly 8 makes out of 10 is about 30.2%.

Common Mistakes & Tips

  • !Applying the binomial model when trials are not independent (e.g., sampling without replacement from a small population; use hypergeometric instead).
  • !Confusing P(X = k) with P(X >= k); the latter requires summing from k to n.
  • !Using the normal approximation when np or n(1-p) is less than 5, which gives inaccurate results.

Related Concepts

Poisson Distribution

Approximates the binomial when n is large and p is small, modeling rare events.

Combination

The binomial coefficient C(n,k) is essential for computing binomial probabilities.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Coin Flip Probability Calculator

Combination Calculator

Poisson Distribution Calculator

Frequently Asked Questions

What are the assumptions of the binomial distribution?

Fixed number of trials n, each trial has only two outcomes (success/failure), constant probability p across trials, and trials are independent.

When should I use Poisson instead of binomial?

Use Poisson when n is very large (>100) and p is very small (<0.01), making np moderate. The Poisson distribution with lambda = np provides a good approximation.

How do I calculate cumulative binomial probability?

P(X <= k) = sum of P(X=i) for i from 0 to k. Most statistical software and calculators have a built-in binomial CDF function.