Binomial probability
Find exactly $k$ successes in $n$ independent Bernoulli trials.
Binomial probability is one of 7 distributions formulas in the probability section of this library, and it is used at high school · ap · university level.
Why binomial probability works
Any one specific sequence with k successes and the rest failures has probability p to the k times the failure probability to the remaining power, because the trials are independent. Then the coefficient counts how many different orderings produce exactly k successes, and those orderings are mutually exclusive, so their probabilities add.
What each symbol means
$p$ is constant success probability.
Binomial probability: when it holds
Fixed $n$, independent trials, two outcomes, and constant $p$.
When it stops applying
It needs a trial count fixed in advance, independent trials, exactly two outcomes, and a success probability that never changes. Drawing 5 cards from a deck breaks the last two: it predicts 0.3125 for exactly two red cards, while the correct hypergeometric answer is about 0.3251.
Binomial probability: a worked example
For $n=4,p=.5$, $P(X=2)=\binom42(.5)^4=6/16$.
The mistake to avoid
What people do: Omitting the coefficient and reporting only the probability of one particular sequence.
Why it goes wrong: For four fair coin flips, exactly two heads has probability 6 times one sixteenth, which is 0.375. Leaving out the 6 gives 0.0625, the chance of one specific pattern such as heads, heads, tails, tails.
Do this instead: Count the orderings first with a combination, then multiply that count by the probability of a single sequence.
Binomial probability: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $p$ is constant success probability.
- Check the conditions before substituting. Fixed $n$, independent trials, two outcomes, and constant $p$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most probability slips.
Where this formula fits
- Subject
- Probability formulas — 17 entries in this library
- Topic
- Distributions
- Level
- High school · AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where binomial probability comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Probability Foundations — the lesson behind this formula: model outcomes, complements, and conditional information.
- Statistics Calculator — check your substitution and the value it produces.
- Study probability — the subject guide that explains the ideas these formulas compress.
- Statistics Practice — questions that make you retrieve the formula instead of recognising it.
- All 17 probability formulas — the full grouped reference, or the complete formula library.
Questions about binomial probability
How do I handle at least k successes?
Add the individual probabilities from k upward, or subtract the probabilities below k from 1. The complement route is usually shorter when k is small, such as at least one.
Does the number of trials really have to be fixed ahead of time?
Yes. If you keep going until the first success, the trial count is random and the geometric distribution applies instead, with a different formula and a different mean.
Can the success probability change between trials?
Not for this formula. A changing probability, such as a player who tires over a long match, breaks the model even if every other condition holds.
How do I compute this without doing the arithmetic by hand?
Use the built-in binomial functions, called binompdf for an exact count and binomcdf for a running total. They handle large factorials that would otherwise overflow.