Question

1. t - shirt factories two t - shirt printing factories produce t - shirts for a local sports team. factory a produces 60% of the shirts and factory b produces 40%. five percent of the shirts from factory a are defective, and 6% of the shirts from factory b are defective. choose 1 shirt at random. given that the shirt is defective, what is the probability that it came from factory a?

Explanation:

Step1: Calculate the probability of a defective shirt from Factory A

The probability that a shirt is from Factory A, \(P(A)=0.6\). The probability that a shirt from Factory A is defective, \(P(D|A) = 0.05\). So the probability that a shirt is from Factory A and is defective is \(P(A\cap D)=P(A)\times P(D|A)=0.6\times0.05 = 0.03\)

Step2: Calculate the probability of a defective shirt from Factory B

The probability that a shirt is from Factory B, \(P(B)=0.4\). The probability that a shirt from Factory B is defective, \(P(D|B)=0.06\). So the probability that a shirt is from Factory B and is defective is \(P(B\cap D)=P(B)\times P(D|B)=0.4\times0.06=0.024\)

Step3: Calculate the total probability of a defective shirt

Using the law of total probability, \(P(D)=P(A\cap D)+P(B\cap D)=0.03 + 0.024=0.054\)

Step4: Apply Bayes' theorem

By Bayes' theorem \(P(A|D)=\frac{P(A\cap D)}{P(D)}\). Substitute the values we found: \(P(A|D)=\frac{0.03}{0.054}=\frac{30}{54}=\frac{5}{9}\approx0.556\)

Answer:

\(\frac{5}{9}\approx0.556\)