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Binomial Probability Calculator: Exact and Cumulative

This free binomial calculator finds the chance of exactly k successes in n independent trials, as well as the chance of at most or at least k. It also gives the mean, variance and standard deviation.

By Debmedia Technologies LLP · Kolkata, India · Updated

What Is It and When Do You Need It?

The binomial distribution counts successes in repeated yes-or-no trials. The number of heads in 10 coin flips, defective items in a batch of 50 and correct answers on a guessing quiz all follow it.

Enter n, p and k to see the exact and cumulative probabilities.

How to Use This Tool

  1. Enter the number of trials.
  2. Enter the success probability as a decimal or a percentage.
  3. Enter k.
  4. Press Calculate.

How the Calculation Works

The exact probability is C(n, k) p^k (1 − p)^(n − k), computed in log space with the log-gamma function for accuracy. The cumulative values add the exact probabilities for all k up to or from the chosen count. Mean, variance and standard deviation use n p, n p (1 − p) and their square root.

Worked Example

For 10 fair coin flips, the chance of exactly 5 heads is 0.2461, the chance of at most 5 is 0.6230 and the chance of at least 5 is 0.6230.

Common Uses

  • Quality control.
  • Coin and dice problems.
  • Quiz guessing odds.
  • Statistics classes.

Tips and Common Mistakes

  • k cannot be bigger than n.
  • Use the normal distribution for very large n if you only need an approximation.
  • Check that trials are independent.

Next, you may need the probability calculator, permutation and combination calculator, normal distribution calculator, sample size calculator.

Everything runs in your browser. Nothing you type is uploaded, stored or sent to any outside service.

Frequently Asked Questions

When you repeat the same trial n times, each with two outcomes, with the same chance of success and independent results, and count the successes.
P(X = k) = C(n, k) × p^k × (1 − p)^(n − k), where C(n, k) is the number of ways to choose k from n.
Mean = n × p and standard deviation = √(n × p × (1 − p)).
Up to 10,000. The tool uses logarithms, so large values do not overflow.

Disclaimer: results are calculated in your browser from the numbers you enter. Check important results before you rely on them for money, health, safety or legal decisions.