Theory question 112
ProbabilityIn an examination, 60% of the candidates passed. If 10 candidates are selected at random, find the probability that: (a) at least two of them failed; (b) exactly half of them passed; (c) at most two of them failed.
Show the model answer
Model answer
Binomial: n = 10, P(fail) = 0.4, P(pass) = 0.6.
(a) 1 − 0.6¹⁰ − 10(0.4)(0.6)⁹ ≈ 1 − 0.0060 − 0.0403 = 0.9536.
(b) ¹⁰C₅(0.6)⁵(0.4)⁵ ≈ 0.2007.
(c) P(0) + P(1) + P(2) = 0.0060 + 0.0403 + 0.1209 ≈ 0.1673.
Explanation
Use P(X = r) = ⁿCᵣ pʳ qⁿ⁻ʳ with p = probability of failing.