Given that the item is defective what is the chance that

Question

Suppose that the birthday of each of three people is equally likely to be any one of the 365 days of the year, independently of others. Let Bij denote the event that person i  has the same birthday as person j, where the labels i and j may be 1. 2, or 3.

a) Are the events B₁₂ and B₂₃ independent?

b) Are the events B₁₂, and B₁₃ independent?

c) Are the events B₁₂, B₂₃, and B₁₃ pairwise independent?

Q1: A factory produces items in boxes of 2. Over the long run:

92% of boxes contain 0 defective items:

5% of boxes contain 1 defective item and

3% of boxes contain 2 defective items,

A box is picked at random from production, then an item is picked at random from the box. Given that the item is defective, what is the chance that the second item in the box is defective?