The probability that a person succeeds to pick the longest stick from a randomly ordered n sticks of distinct lengths following the optimal strategy?
If you tried to get the longest stick out of 10, and they were shown to you in random order, how would you proceed?
Here’s a simple strategy, based on the simple idea that it’s daft to pick a stick that isn’t the longest so far: Don’t take any of the first four sticks, then pick the first stick that is longer than the longest so far. How will this work out?
If the first four contain the longest stick, you lose. If the first four contain the second longest stick, you win because you will pick the longest. If the first four contain the third longest stick, then you will pick the longest or second longest, 50% chance to win, and so on.
Chances are 65.61% that the first four don’t contain the longest, and 40.96% that they contain neither of the two longest; that alone gives you a 24.65% chance to win because you must pick the longest stick. You can calculate your chances to win quite easily: let p(k) be probability (“None of the p largest”) - probability (“None of the p+1 largest”), then your chance to win is p(1) + p(2)/2 + p(3)/3 ...
You would also check if skipping the first three or five or any other number might be better.
If you ignore about the first n/e sticks, this will give you a winning chance around 1/e even when n is very large.