Theory question 113
Probability(a) 12% of sugar packaged by a machine in a factory are found to be less than 50kg. If 8 of the bags were selected at random, find, correct to 4 significant figures, the probability that: (i) exactly half; (ii) at least 75% of the bags of sugar are greater than or equal to 50kg
(b) There are 35% defective syringes in a large box. If 14 syringes are selected at random from the box, calculate the probability that between three and six are defective.
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(a) p(≥ 50 kg) = 0.88, q = 0.12, n = 8.
(i) P(X = 4) = ⁸C₄(0.88)⁴(0.12)⁴ = 70 × 0.59970 × 0.00020736 ≈ 0.008705.
(ii) At least 6 of 8: P(6) + P(7) + P(8) = 28(0.88)⁶(0.12)² + 8(0.88)⁷(0.12) + (0.88)⁸ = 0.18725 + 0.39233 + 0.35963 ≈ 0.9392.
(b) p = 0.35, q = 0.65, n = 14. Between 3 and 6 (that is, 4 or 5): P(4) + P(5) = 1001(0.35)⁴(0.65)¹⁰ + 2002(0.35)⁵(0.65)⁹ = 0.2022 + 0.2178 ≈ 0.4200.
(If 3 and 6 are included: add P(3) = 0.1366 and P(6) = 0.1759, giving 0.7325.)
Use the binomial distribution P(X = r) = ⁿCᵣ pʳ qⁿ⁻ʳ.