Question

A computer manufacturer uses chips from three sources. chips from sources a, b and c are defective with probabilities 0.005, 0.001 and 0.01, respectively. the proportions of chips from a, b and c are 0.5, 0.1 and 0.4 respectively. if a randomly selected chip is found to be defective, find (a) the probability that the chips are from a. (b) the probability that the chips are from b. (c) the probability that the chips are from c

Answer 1

To find the probabilities that a randomly selected chip is from each source, we can use Bayes' theorem. Let's define the events: A, B, C, D. We are given the probabilities of the chip being defective given the source, as well as the proportions of chips from each source. We can calculate the probabilities using Bayes' theorem and the law of total probability.

Step-by-step explanation:

To find the probabilities that a randomly selected chip is from each source, we can use Bayes' theorem. Let's define the events:

• A = the chip is from source A
• B = the chip is from source B
• C = the chip is from source C
• D = the chip is defective

We are given:

• P(D|A) = 0.005
• P(D|B) = 0.001
• P(D|C) = 0.01
• P(A) = 0.5
• P(B) = 0.1
• P(C) = 0.4

(a) To find the probability that the chip is from source A given that it is defective, we need to find P(A|D). Using Bayes' theorem:

P(A|D) = (P(D|A) * P(A)) / P(D) = (0.005 * 0.5) / P(D)

Similarly, we can calculate P(B|D) and P(C|D) using the same formula. Finally, we can calculate P(D) using the law of total probability:

P(D) = P(D|A) * P(A) + P(D|B) * P(B) + P(D|C) * P(C)