Consider a machine that produces items in sequence. Under normal operating conditions, the items are independent with probability 0.01 of being defective. However, it is possible for the machine to develop a “memory” in the following sense: After each defective item, and independent of anything that happened earlier, the probability that the next item is defective is 2/5. After each nondefective item, and independent of anything that happened earlier, the probability that the next item is defective is 1/165. Assume that the machine is either operating normally for the whole time we observe or has a memory for the whole time that we observe. Let B be the event that the machine is operating normally, and assume that Pr(B) = 2/3. Let Di be the event that the ith item inspected is defective. Assume that D1 is independent of B.
a. Prove that Pr(Di) = 0.01 for all i. Use induction.
b. Assume that we observe the first six items and the event that occurs is E = Dc1 ∩ Dc2 ∩ D3 ∩ D4 ∩ Dc5 ∩ Dc6. That is, the third and fourth items are defective, but the other four are not. Compute Pr(B|D).