If X ~ B(n, p), the mean is E(X) = np and the variance is Var(X) = np(1 − p). For X ~ B(7, 0.9), the mean is 6.3 and the variance is 7 × 0.9 × 0.1 = 0.63.
Cumulative probabilities add individual ones: P(X ≤ 2) = P(0) + P(1) + P(2). Use the complement for 'at least': P(X ≥ 3) = 1 − P(X ≤ 2). Statements like 'fewer than 4' mean X ≤ 3, so translate the words carefully.
Worked example
X ~ B(11, 0.2). Find P(X = 1) to 3 s.f.
- P(X = 1) = ¹¹C₁ × 0.2 × 0.8¹⁰.
- 11 × 0.2 × 0.10737 = 0.2362…
Answer:0.236
Key words
- E(X)
- the expected value, np
- Var(X)
- the variance, np(1 − p)
- complement
- 1 minus the probability
Practise this
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Common Core-referenced: HSS-MD.A.3