A manufacturer of automotive g
A manufacturer of automotive gaskets has two plants, A and B.Plant A manufactures 65 % of the gaskets and plant B manufactures35 %. Because of a batch of faulty material from a companysupplying both plants, 3% of the gaskets are of sub-standardquality from plant A and 5% are sub-standard from plant B. Despiteyour internal quality control procedures, you still had asub-standard gasket returned from one of your customers. What isthe probability it came from plant B? i) Illustrate your answerwith an event tree and ii) show how to calculate the value usingBayes Equation. b) Using Bayes Theorem, solve the following twoproblems: 1. A couple has three children, the eldest of which is aboy. What is the probability that they have three boys? 2. A couplehas three children, one of which is a boy. What is the probabilitythat they have three boys?
Answer:
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 eldestis boy , P(E | 3B) = 1
Given, the eldest of which is a boy. Probability that they havethree 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 isboy , P(B | 3B) = 1
Given, the one of children is a boy. Probability that they havethree boys = P(3B | B)
= (B | 3B) P(3B) / P(B) {By Bayes theorem}
= 1 * (1/8) / (7/8)
= 1/7
"Our Prices Start at $11.99. As Our First Client, Use Coupon Code GET15 to claim 15% Discount This Month!!"
