A personnel manager is to interview four candidates for a job. These are ranked 1, 2, 3, and 4 in order of preference and will be interviewed in random order. However, at the conclusion of each interview, the manager will know only how the current candidate compares to those previously interviewed. For example, the interview order 3, 4, 1, 2 generates no information after the first interview, shows that the second candidate is worse than the first, and that the third is better than the first two. However, the order 3, 4, 2, 1 would generate the same information after each of the first three interviews. The manager wants to hire the best candidate but must make an irrevocable hire/no hire decision after each interview. Consider the following strategy: Automatically reject the first ?s ?candidates and then hire the first subsequent candidate who is best among those already interviewed (if no such candidate appears, the last one interviewed is hired). For example, with s =2 , the order 3, 4, 1, 2 would result in the best being hired, whereas the order 3, 1, 2, 4 would not. Of the four possible ?s ?values (0, 1, 2, and 3), which one maximizes ?P?(best is hired)? [?Hint: ?Write out the 24 equally likely interview orderings: s=0 means that the first candidate is automatically hired.]

Problem 111E Answer: Step1: We have A personnel manager is to interview four candidates for a job. These are ranked 1, 2, 3, and 4 in order of preference and will be interviewed in random order. However, at the conclusion of each interview, the manager will know only how the current candidate compares to those previously interviewed. For example, the interview order 3, 4, 1, 2 generates no information after the first interview, shows that the second candidate is worse than the first, and that the third is better than the first two. However, the order 3, 4, 2, 1 would generate the same information after each of the first three interviews. The manager wants to hire the best candidate but must make an irrevocable hire/no hire decision after each interview. Consider the following strategy: Automatically reject the first s candidates and then hire the first subsequent candidate who is best among those already interviewed (if no such candidate appears, the last one interviewed is hired). For example, with s =2 , the order 3, 4, 1, 2 would result in the best being hired, whereas the order 3, 1, 2, 4 would not. Of the four possible s values (0, 1, 2, and 3). We need to find which one maximizes P (best is hired) Step2: 1).Consider the order (1,2,3,4) Here the last number is 4 hence, n = 4 Now, Apply the permutation to the above given order for s = 0,1,2,3. That is for s = 0, P s P 0 = 1 For s = 1, P s P 1 = 4 For s = 2, Ps P 2 = 12 = 4 (we have n = 4 so, in order number 12 is 4) n 4 For s = 3, P s P 3 = 24 = 4 (we have n = 4 so, in order number 24 is 4) 2).Consider the order(1,2,4,3) Here the last number is 3 hence, n = 3 Now, Apply the permutation to the above given order for s = 0,1,2,3. n 3 That is for s = 0, s = 0 P = 1 n 3 For s = 1,s = 1P = 3 n 3 For s = 2, s = 2 P = 6 = 3 (we have n = 3 so, in order number 6 is 3) For s = 3, s = 3 = 6 = 3 (we have n = 3 so, in order number 6 is 3) Similarly, For other orders can shown in the following table permutations S = 0 S = 1 S = 2 S = 3 (1,2,3,4) 1 4 4 4 (1,2,4,3) 1 3 3 3 (1,3,2,4) 1 4 4 4 (1,3,4,2) 1 2 2 2 (1,4,2,3) 1 3 3 3 (1,4,3,2) 1 2 2 2 Step3: Consider, permutations S = 0 S = 1 S = 2 S = 3 (2,1,3,4) 2 1 4 4 (2,1,4,3) 2 1 3 3 (2,3,1,4) 2 1 1 4 (2,3,4,1) 2 1 1 1 (2,4,1,3) 2 1 1 3 (2,4,3,1) 2 1 1 1 Again consider, permutations S = 0 S = 1 S = 2 S = 3 (3,1,2,4) 3 1 4 4 (3,1,4,2) 3 1 2 2 (3,2,1,4) 3 2 1 4 (3,2,4,1) 3 2 1 1 (3,4,1,2) 3 1 1 2 (3,4,2,1) 3 3 2 1 Step4: Consider, permutations S = 0 S = 1 S = 2 S = 3 (4,1,2,3) 4 1 3 3 (4,1,3,2) 4 1 2 2 (4,2,1,3) 4 2 1 3 (4,2,3,1) 4 2 1 1 (4,3,1,2) 4 3 1 2 (4,3,2,1) 4 3 2 1 Since,there are 24 total outcomes for each value of s. P(Hire 1/s = i) 6/24 11/24 10/24 6/24 Therefore,the probability that the best is hired is s =1.