Question

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.

(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?

Answer 1

Probability that a batch is defective = 1% that is p=0.01 Pl at least 2 out of 100 circuit boards defective) = P(at least I out of 50 batches defective) P( me batches defective out of 50) = 50, þ° (1-)90 = (1-6350 = (0-99)50 = 0.605006 20-605 Plat least 1 out of 50 batches defectine). = 1-PC no batches defective out of 50). = 1-0.605006 = 0.39499 ~0.395 Expected number of defective batches out of 50 = 50! = 50 X0.01 = 0.5 defective circuit boards = 0.582 - Expected number of

when each circuit board is made seperately PC at least 2 out of 100 circuite boards defective) = 1 - P (no circuit boards defective) - P(exactly on circuit bodo defective) = 1- 100 po (i-poo 100c, p' (1-6,99 = 1- (0.99000 - 100 (0.01) (0.99,99 = 1 -0.3660 -0.3697 = 0.2642 Expected number of defective boards out of 100 = 100 p = 100 X0.01 = 1 Although expected number of defective boards is same in both Cases, the probability of having at least 2 defective products is more when the circuit boards are made in a batch of 2.