Binomial Probability Calculator
Compute the exact probability of getting exactly k successes in n independent trials, each with probability p. Use this when modeling coin flips, quality control tests, or any repeated yes/no experiment.
About this calculator
The binomial probability formula gives the chance of exactly k successes in n independent trials: P(X = k) = C(n,k) × p^k × (1−p)^(n−k), where C(n,k) = n! / (k! × (n−k)!) is the binomial coefficient. Each trial must be independent and have the same probability p of success. The mean (expected value) of a binomial distribution is μ = n × p, and the variance is σ² = n × p × (1−p). Cumulative probabilities sum individual P(X = k) values from 0 up to k. This distribution is foundational in statistics, genetics, quality assurance, and risk analysis.
How to use
Suppose you flip a fair coin 10 times and want the probability of exactly 6 heads. Set n = 10, k = 6, p = 0.5. First compute C(10,6) = 210. Then: P = 210 × (0.5)^6 × (0.5)^4 = 210 × 0.015625 × 0.0625 = 210 × 0.000977 ≈ 0.2051. So there is about a 20.5% chance of getting exactly 6 heads in 10 fair coin flips.
Frequently asked questions
What is the difference between exact and cumulative binomial probability?
Exact binomial probability gives the chance of getting precisely k successes, such as exactly 3 out of 10. Cumulative probability sums the probabilities for all values up to k, answering questions like 'at most 3 successes.' You can also compute upper-tail probability for 'at least k successes' by subtracting the cumulative value from 1. Choosing the right type depends on the question you are trying to answer.
When should I use a binomial distribution instead of a normal distribution?
Use the binomial distribution when trials are discrete, countable, and each has only two outcomes (success or failure). The normal distribution is a continuous approximation that works well when n is large and p is not too close to 0 or 1, typically when n×p ≥ 5 and n×(1−p) ≥ 5. For small n or extreme p values, stick with the exact binomial formula for accuracy. In quality control with small sample sizes, the binomial is almost always preferred.
How does changing the probability of success affect the binomial distribution shape?
When p = 0.5 the binomial distribution is perfectly symmetric around its mean n×p. As p moves toward 0 the distribution skews right, concentrating probability near zero successes. As p approaches 1 it skews left, clustering near n successes. Increasing n makes the distribution more bell-shaped regardless of p, due to the central limit theorem. Visualizing this helps in understanding risk and rare-event modeling.