(mathematics) Persian Princess selected bridegroom

Topic:

Persian princess to the marriageable age, to choose Bridegroom. There are 100 candidates, all of whom the princess has not seen. hundred people in random order, from the princess in front of each passing. Whenever a man passes in front of the princess, the princess either chose him as a bridegroom or not. If he had been chosen, the rest of the men who had not yet appeared would have been dismissed from their homes, and the campaign for bridegroom had ended. If not, the current man will leave, that is, pass this person, the next person to debut. The princess is not allowed to go back and choose from the pass-off. The rule is that the Princess must elect a person in this hundred to do bridegroom, that is to say, if the first 99 people princess can not see, she must choose the 100th man for bridegroom, no matter how ugly he is.

The task is to design a method for the princess, so that she has the highest probability of choosing the most handsome men of the Hundred bridegroom.

Ideas:

The point is, there is no choice to ensure that the princess must choose the most handsome handsome guy. For any selection method, there are always some appearances in order to let the princess and the handsome miss. Therefore, the question is not to win the election method (because it does not exist), but the highest probability of the election method.

First answer: first rejected the 100/e man (the natural logarithm of E, 2.718...,100/e about 37), remember the most handsome of the 37 men, and then in the back of the men, find the first more handsome than the previous 37 handsome man, that is bridegroom. If not met, then had to choose a 100th man.

Abstract as a mathematical model: the first to reject the K-person, and then in the back of the people to find better.

Theoretical deduction: How to find the best k?

For a fixed k, if the most suitable person appears in the first position (K < i≤n), in order to let him lucky just by the MM selected, you have to meet the former i-1 personal best person in the former K person, this has k/(i-1) possible. Considering all the possible I, we get the total probability P (k) of the best boys to be selected after the first K boys are tested:

By using X to represent the value of the k/n, and assuming that N is sufficiently large, the above formula can be written as:

To-x ln x, which is a derivative of 0, can solve the optimal value of x, which is the reciprocal--1/e of the mysterious constant of Euler's study!

This conclusion can be tested by the 37%-rule simulation experiment to see how large the probability of selection, and then post the code.

Resources:

http://www.e-future.com.cn/news_details.php?nid=1702

http://songshuhui.net/archives/57722

　　

(mathematics) Persian Princess selected bridegroom