Question

Exercise 12.8.1: Probability of manufacturing defects. The probability that a circuit board produced by a particular manufacturer has a defect is 1%. You can assume that errors are independent, so the event that one circuit board has a defect is independent of whether a different circuit board has a defect. (a) What is the probability that out of 100 circuit boards made exactly 2 have defects? (b) What is the probability that out of 100 circuit boards made at least 2 have defects? (c) What is the expected number of circuit boards with defects out of the 100 made? (d) Now suppose that the circuit boards are made in batches of two. Either both circuit boards in a batch have a defect or they are both free of defects. The probability that a batch has a defect is 1%. What is the probability that out of 100 circuit boards (50 batches) at least 2 have defects? What is the expected number of circuit boards with defects out of the 100 made? How do your answers compared to the situation in which each circuit board is made separately? PLEASE ANSWER a) through c) explaining each step by step and legible

Answer 1

a. Given,

n=100, p(defective)=0.01 , p(non-defective)=1-0.01=0.99

Probability of having exactly three defects = P(X=2) = 100C2 x (0.01)^2 x (0.99)^98

= 4950 x 0.0001 x 0.373

= 0.184653

b. Given,

n=100, p(defective)=0.01 , p(non-defective)=0.99

Probability of atleast two defective can be calculated as P(atleast one defective)-P(exactly one defective)

P(atleast one defective)= 1 - P(none defective)

=1 - (0.99)^100 = 1-0.366 =0.634

P(exactly one defective) = P(X=1) = 100C1 x (0.01)^1 x (0.99)^99

=100 x 0.01 x 0.369 = 0.369

Therefore, P(atleast three defective) = 0.634-0.369 = 0.265

c. The percentage of defective items is 1% i.e 1 out of 100 circuit boards would be defective. Therefore, the expected number of defects out of 100 circuit boards would be 1.