Question

Discrete Structures Mathematics

Two different factories both produce a certain automobile part. The probability that a component from the first factory is defective is 4%, and the probability that a component from the second factory is defective is 5%. In a supply of 160 of the parts, 100 were obtained from the first factory and 60 from the second factory.

(a) What is the probability that a part chosen at random from the 160 is from the first factory?

(b) What is the probability that a part chosen at random from the 160 is from the second factory?

(c) What is the probability that a part chosen at random from the 160 is defective?

(d) If the chosen part is defective, what is the probability that it came from the first factory?

Answer 1

a) 62.5%

b) 37.5%

c) 4.3%

d) 71%

Explanation:

A=1: component from the 1st factory

A=2: component from the 2nd factory

X=0: component OK

X=1: component defective

a) P(A=1)=100/160=0.625=62.5%

b) P(A=2)=60/160=0.375=37.5%

c) P(X=1) = P(X=1 & A=1) + P(X=1 & A=2) = P(X=1|A=1).P(A=1) + P(X=1|A=2).P(A=2) = 0.04*100/160 + 0.05*60/160 = 0.025 + 0.1875 = 0.04375 = 4.3%

d) Based on Bayes’ theorem:

P(A=1|X=1) = P(X=1|A=1).P(A=1)/P(X=1) = 0.05*(100/160)/0.04375 = 5/7 = 0.71 = 71%