Binomial Probability

Calculate exact and cumulative probabilities for the number of successes in n independent trials, each with success probability p. Works for large n (up to 100 000) without overflow.

StatisticsDiscreteProbability

Inputs

Input examples

Calculate

Click Solve next to the quantity you want to find: it moves into the result below. Then fill in the others. Example: P(X ≥ k) with k = 1, p = 1/6 and probability 0.95 — solve n for the number of dice rolls needed.

Number of trials n

Success probability p

Number of successes k

Probability P(X = k)

What is a Binomial Distribution?

A binomial experiment repeats the same yes/no trial n times, independently, with the same success probability p each time — like flipping a coin 10 times and counting heads. X, the number of successes, follows the binomial distribution.

The coefficient C(n, k) counts in how many orders the k successes can occur; p^k(1−p)^n−k is the probability of any one such order.

Meanμ = np

Varianceσ² = np(1−p)

p = 0.5distribution is symmetric

np ≥ 5approx. normal

Live Distribution

B(10, 0.5)

selected outcomes

other outcomes

normal approximation

Result

Enter the parameters above

Distribution Properties

Property	Formula

Key Techniques

Binomial Coefficient

The number of ways to choose which k of the n trials are successes. These are the numbers in Pascal’s triangle.

Example:

Complement Rule

“At least one” questions are easiest through the complement: one minus the probability of no successes at all.

Example:

Normal Approximation

For large n (np and n(1−p) both ≥ 5) the bars follow a bell curve. The +0.5 continuity correction accounts for the bar width.

Example:

Sources & Verification

References

S. M. Ross, A First Course in Probability, 10th ed., Pearson, 2019 — Sec. 4.6 (The Bernoulli and Binomial Random Variables).
D. C. Montgomery & G. C. Runger, Applied Statistics and Probability for Engineers, 6th ed., Wiley, 2013 — Ch. 3 (Discrete Random Variables and Probability Distributions).