X ~ B(n, p) models the number of successes in n independent trials, each with probability p of success. P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ.
For X ~ B(11, 0.15), P(X = 3) = ¹¹C₃ × 0.15³ × 0.85⁸ = 165 × 0.003375 × 0.2725 = 0.152. Cumulative probabilities like P(X ≤ 2) add the separate probabilities, and P(X ≥ 1) = 1 − P(X = 0).
The mean of a binomial distribution is np, which is the number of successes you would expect on average.
Worked example
X ~ B(5, 0.5). Find P(X = 2).
- ⁵C₂ = 10.
- 10 × 0.5² × 0.5³ = 10 × 0.03125.
Answer:0.3125
Key words
- trial
- one attempt with two outcomes
- success
- the outcome being counted
- B(n, p)
- a binomial distribution
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Common Core-referenced: HSS-MD.A.3