Binomial probability calculator

Formula

How it works

Each of the n trials succeeds with probability p and fails with probability 1 − p. The number of ways to get exactly k successes is nCk, and each specific arrangement with k successes has probability pᵏ(1 − p)ⁿ⁻ᵏ.

Worked example

The same calculation with the default values:

• Number of trials (n) 10
• Probability of success 50 %
• Number of successes (k) 4

1. P(X = 4) = C(10,4) × 0.5^4 × (1 − 0.5)^(10 − 4) = 20.507812%
2. mean = 10 × 0.5 = 5
3. standard deviation = √(10 × 0.5 × (1 − 0.5)) = 1.581139

Result: P(X = k): 20.51% · P(X ≤ k): 37.7% · Mean (expected successes): 5

Common mistakes

• Using this formula for a situation where trials aren't independent or the success probability changes between trials - the binomial model strictly requires a fixed number of independent trials each with the same probability of success.
• Confusing P(X = k) (exactly k successes) with P(X ≤ k) or P(X ≥ k) (at most or at least k successes) - these are different calculations; this calculator's formula computes the exact-k case specifically.
• Entering a probability as a percentage (70) instead of a decimal (0.7) - the formula expects p between 0 and 1.