BdMO 2016 national

For students of class 9-10 (age 14-16)
soyeb pervez jim
Posts: 21
Joined: Sat Jan 28, 2017 11:06 pm

BdMO 2016 national

Unread post by soyeb pervez jim » Wed Mar 28, 2018 8:03 pm

Juli is a mathematician and devised an algorithm to find a husband. The strategy is:
• Start interviewing a maximum of $1000$ prospective husbands. Assign a ranking $r$ to each person that is a positive integer. No two prospects will have same the rank $r$.
• Reject the first $k$ men and let $H$ be highest rank of these $k$ men.
• After rejecting the first $k$ men, select the next prospect with a rank greater than $H$ and then stop the search immediately. If no candidate is selected after $999$ interviews, the $1000th$ person is selected.

Juli wants to find the value of $k$ for which she has the highest probability of choosing the highest ranking prospect among all $1000$ candidates without having to interview all $1000$ prospects.
(a) (6 points:) What is the probability that the highest ranking prospect among all $1000$ prospects is the $(m + 1)th$ prospect?
(b) (6 points:) Assume the highest ranking prospect is the $(m + 1)th$ person to be interviewed. What is the probability that the highest rank candidate among the first $m$ candidates is one of the first $k$ candidates who were rejected?
(c) (6 points:) What is the probability that the prospect with the highest rank is the $(m+1)th$ person and that Juli will choose the $(m+1)th$ man using this algorithm?
(d) (16 points:) The total probability that Juli will choose the highest ranking prospect among the $1000$ prospects is the sum of the probability for each possible value of $m+1$ with $m+1$ ranging between $k+1$ and $1000$.
Find the sum. To simplify your answer use the formula
$In N \approx \frac{1}{N-1}+\frac{1}{N-2}+...+\frac{1}{2}+1$
(e) (6 points:) Find that value of $k$ that maximizes the probability of choosing the highest ranking prospect without interviewing all $1000$ candidates. You may need to know that the maximum of the function $x ln \frac{A}{x-1}$ is approximately $\frac{A + 1}{e}$, where $A$ is a constant and $e$ is Euler’s number, $e = 2.718....$

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samiul_samin
Posts: 1004
Joined: Sat Dec 09, 2017 1:32 pm

Re: BdMO 2016 national

Unread post by samiul_samin » Fri Mar 30, 2018 11:30 am

This problem is solved here.

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