The binomial formula
P(X = k) = C(n,k) · pk · (1−p)n−k
It has three parts, and each does a specific job.
pk(1−p)n−k is the probability of one particular sequence with k successes and n−k failures, while C(n,k) counts how many different sequences can contain those k successes. Multiply them together.
When the binomial distribution applies
Four conditions, all of which must hold. Most wrong answers come from using the formula where one of them does not.
- A fixed number of trials, decided before you start.
- Two outcomes per trial — success or failure, however you define them.
- The same probability every trial.
- Independent trials: one result tells you nothing about the next.
A common reason the binomial model does not apply is the third condition: sampling without replacement. Drawing 5 cards from a deck without replacement is not binomial, because the probability shifts with every card removed. Drawing 5 times with replacement is.
Exactly, at most, at least: the part that costs marks
Binomial questions are usually lost in reading, not arithmetic. These are five different quantities:
| Wording | Notation | Includes k? |
|---|---|---|
| Exactly k | P(X = k) | Only k |
| Fewer than k | P(X < k) | No |
| At most k, no more than k | P(X ≤ k) | Yes |
| At least k, k or more | P(X ≥ k) | Yes |
| More than k | P(X > k) | No |
“At least k” and “more than k” differ by exactly
P(X = k), which is why this calculator shows all five
together rather than making you pick one and hope.
Mean, variance and standard deviation
Mean = np Variance = np(1−p) SD = √(np(1−p))
For 10 coin flips: the mean is 5, the variance 2.5, and the standard
deviation √10/2, about 1.58. The mean is where the
distribution balances — and like any expected value it need not be
a whole number, so 10 trials at p = 0.37 have a mean of 3.7 successes.
Why the answers here are exact
Binomial probabilities are rational numbers, and this calculator keeps
them that way: every term shares the denominator
bn, so the whole distribution is a set of whole
numbers over one common bottom. Cumulative probabilities are then
additions of integers rather than a chain of roundings, which is where
floating-point versions lose accuracy on long tails.
Exact fractions get unwieldy quickly — P(X=7) for n=20 and p=0.37 has a 40-digit denominator — so the decimal leads and the fraction is shown only when it is short enough to be read.
Frequently asked questions
What is binomial probability?
The probability of getting exactly k successes in n independent trials, where each trial has the same probability p of success. The formula is P(X = k) = C(n,k) · p^k · (1−p)^(n−k).
What does C(n,k) mean?
The number of ways to choose k items from n without regard to order — also written nCk or "n choose k". In the binomial formula it counts the different orders in which the k successes could occur.
What is the difference between "at least" and "more than"?
"At least k" includes k itself; "more than k" does not. The two differ by exactly P(X = k). The same distinction separates "at most" from "fewer than".
When should I not use the binomial distribution?
When the trials are not independent or the probability changes between them. Sampling without replacement is the usual case — drawing cards from a deck without putting them back needs the hypergeometric distribution instead.
Can p be 0 or 1?
Yes. Both are valid, if degenerate: with p = 0 every trial fails and all the probability sits on k = 0, and with p = 1 every trial succeeds. The calculator handles both rather than rejecting them.