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?

Question

Mathematics

Lightning

24 January, 05:42

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?

+1

Answers (1)

Know the Answer?

Branden Barker · Answer 1

a) 62.5%

b) 37.5%

c) 4.3%

d) 71%

Step-by-step 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%