This problem sometimes goes by the name of “the secretary problem”. A hat has 10

Question

This problem sometimes goes by the name of “the secretary problem”.

A hat has 100 slips of paper in it with different real numbers written on them. The numbers are all different, but they need not be integers: they could be –17, or 102 , or –22/7. You reach into the hat, pull out slips of paper one at a time, and look at each number. At any point, you may choose to stop. (If you get to the last slip, you stop automatically.) If the last slip you draw has the largest number on it (largest of all 100 numbers, i.e. larger than both the number(s) you have drawn and the numbers that are left in the hat), then you win $10. Otherwise you win nothing.

a) What is a reasonable strategy to use for playing the game? How should you decide when to stop?

b) What is a reasonable amount to pay for the privilege of playing this game? Ten cents? Fifty cents? One dollar? Two dollars?

Hint: You may wat to get started by considering a similar problem in which there are only a small number of slips of paper. Also, it is rather difficult to find an exact solution to the problem, but you should be able to make some estimate of the expected value that is in the right ballpark.

Explanation / Answer

Since the 100 slips of paper can be any number on the real line, and all the hundred numbers are different, the probability of a randomly selected slip being the maximum is 1/100.

a) There is no optimal strategy for the secretary problem. A typical strategy might be to sample until the current maxima is doubled. However, the numbers being randomly choosen from the real line does not guarantee the reasonable-ness of this strategy.

b) Since there is no contraint on the number of slips to be drawn among the 100 slips, the reasonable amount to pay for the privilege of playe this game would be any amount less than the reward $10.

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