Preparing your question to be 56 . t t . Question 1: (15 marks) U 0'5 View question (a) After packing k boxes (numbered 1:2, . . . , k) of 7'\" items each, workers discovered that one defective item has slipped in among the k X m items packed. In an attempt to nd the defective item, they randomly sample n items from each box and examine these. (i) Find the probability that the defective item is in box i. (ii) Find the probability that the defective item is in box 1 , given that it was not found in the sample from box 1. (iii) Find the probility that the defective item is in box 2, given that it was not found in the sample from box 1. (iv) What happens to the probabilities in parts (ii) and (iii) as n w m and as n a 0? (b) A bus arrives at a bus stop at a randomly chosen time within a 1-hour period. A passenger arrives at the bus stop at a randomly chosen time within the same 1-hour period. The passenger is willing to wait for the bus for up to 15 minutes. What is the probability that the passenger will catch the bus? You should assume that the bus arrival time and the passenger arrival time are independent. Hint: You may assume that the two arrival times have a joint distribution that is uni- formly distributed