Rank order is random as given in the question. 4. Mr. Williams is interviewing c

Question

Rank order is random as given in the question.

4. Mr. Williams is interviewing candidates to be his assistant. As he interviews the candidates, he can determine the relative rank of the candidates but not the true rank. For example, if there are six candidates and their true rank is 6, 1, 4, 2, 3, 5, (where 1 is the best) then after he had interviewed the first three candidates he would rank them 3, 1, 2. As he interviews each candidate, he must either accept or reject the candidate. He wants to decide on a strategy for deciding when to stop and accept a candidate that will maximize the probability of getting the best candidate. Assume that there are 2n candidates and they arrive in a random rank order. (a) What is the probability that Mr. Williams gets the best candidate if he interviews all of (b) Assume Mr. Williams decides to interview the first half of the candidates and then Show that he has a better than 25% chance of ending up with the best candidate the candidates? What is it if he chooses the first candidate? continue interviewing until getting a c better than any candidate seen so far

Explanation / Answer

Here N=6,

P=Probablity of selecting one candidate=1/6

q=probablity of not selecting a candidate=1-p

1-1/6=5/6

as per binomial expansion

(q+p)6=q6+6q5p1+15q4p2+20q3p3+15q2p4+6q1p5+p6

=(5/6)6+6.(5/6)5.(1/6)1+15(5/6)4(1/6)2+20(5/6)3.(1/6)3+15(5/6)2(1/6)4+(1/6)6

=0.335+0.402+0.2008+0.0536+.0116+.00002

prob. of at most one success=p0+p1

=.335+.402=.737

prob. of choosing ist candidate=p1=.402

probablity of choosing at least one candidate from them

=1-p0

1-.335=.665

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