Law of Total Probability Exercise

[jinbaw]

Homework Statement

Avril has certain standards for selecting her future husband. She has n suitors and knows how to compare any two and rank them. She decides to date one suitor at a time randomly. When she knows a suitor well enough, she can marry or reject him. If she marries the suitor, she can never know the ones not dated. If she rejects the suitor, she will not be able to reconsider him. In this process, Avril will not choose a suitor she ranks lower than at least one of the previous ones dated. Avril's goal is to maximize the probability of selecting the best suitor. To achieve this, she adopts the following strategy: For some 0 =< m < n, she dumps the first m suitors she dates after knowing each of them well no matter how good they are. Then she marries the first suitor she dates who is better than all those preceding him. In terms of n, find the value of m which maximizes the probability of selecting the best suitor.

Homework Equations

The law of total probability:
{B_1, B_2, ...,B_n} is a partition of the sample space, and A is any event of the sample space. Then, $$P(A) = \sum_{i=1}^{n} P(A | B_{i})P(B_{i})$$

The Attempt at a Solution

I am trying to work on this problem but it gets messy.
The suitors are X1, X2, ... , Xm, X(m+1), ..., Xn.
P(best not in X1, ..., Xm) = 1 - m/n.

Define Bi the event that the best suitor is on the ith spot, and A the event that he is in X1, ..., Xm.
So, $$S = \dot{\bigcup_{i=m+1}}^{n} B_{i} \dot{\cup} A$$ becomes a partition of the sample space. And we know that P(A) = m/n.

We want complement(A) and Bi.
Any help? Is the direction I'm going after correct?

 

[vela]

I think you need to consider where the second-best suitor is located. If he's not among the first m suitors, the first better suitor in the remaining n-m suitors may not be the best.