Question

A manufacturer of automotive gaskets has two plants, A and B. Plant A manufactures 65 % of the gaskets and plant B manufactures 35 %. Because of a batch of faulty material from a company supplying both plants, 3% of the gaskets are of sub-standard quality from plant A and 5% are sub-standard from plant B. Despite your internal quality control procedures, you still had a sub-standard gasket returned from one of your customers. What is the probability it came from plant B? i) Illustrate your answer with an event tree and ii) show how to calculate the value using Bayes Equation. b) Using Bayes Theorem, solve the following two problems: 1. A couple has three children, the eldest of which is a boy. What is the probability that they have three boys? 2. A couple has three children, one of which is a boy. What is the probability that they have three boys?

Answer 1

i)

ii)

Given,

P(A) = 0.65 and P(B) = 0.35

P(S | A) = 0.03 and P(S | B) = 0.05

Probability that sub standard quality came from plant B = P(B | S)

By law of total probability,

P(S) = P(S | A) P(A) + P(S | B) P(B)

= 0.03 * 0.65 + 0.05 * 0.35 = 0.037

By Bayes theorem,

P(B | S) = P(S | B) P(B) / P(S)

= 0.05 * 0.35 / 0.037

= 0.472973

1.

Probability that eldest is boy, P(E) = 1/2

Probability that they have three boys, P(3B) = (1/2)³ = 1/8

Given that all three children are boys, Probability that eldest is boy , P(E | 3B) = 1

Given, the eldest of which is a boy. Probability that they have three boys = P(3B | E)

= (E | 3B) P(3B) / P(E) {By Bayes theorem}

= 1 * (1/8) / (1/2)

= 1/4

2.

Probability that one of three children is boy, P(B) = 1 - Probability that all three are girls = 1 - (1/2)³ = 7/8

Probability that they have three boys, P(3B) = (1/2)³ = 1/8

Given that all three children are boys, Probability that one is boy , P(B | 3B) = 1

Given, the one of children is a boy. Probability that they have three boys = P(3B | B)

= (B | 3B) P(3B) / P(B) {By Bayes theorem}

= 1 * (1/8) / (7/8)

= 1/7