This is exercise 3.13 from the book "elements of analysis and algebra (number theory)" written by Pierre Colmez. The original language is French and below is the translation by Google translate.
In order to test the level of understanding of the 500 members of a same class of the Ecole Polytechnique, they are gathered in the Poincaré amphitheater ; the headmaster says: “I have placed your 500 names in the Arago amphitheater in lockers numbered from 1 to 500, at the equal rate per locker(each person's name is in one locker randomly). I will call you one by one, and ask each one to open lockers one by one looking for your name and then close them without changing the content and return to your room without the possibility of communicating anything to his fellow students remaining in the Poincaré amphitheater. If everyone finds their name in the first 250 lockers they've opened, you can go on vacation. If any of you can't find their name, we'll start over the next day (and I'll change the contents of the lockers of course). You have two hours to design a strategy. » Despair of the students who realize that everyone has a half chance finding their name, and that in total they have $\frac{1}{2^{500}}$ chance of going on vacation after a day, and therefore that they will not go on vacation. Yet after a while, one of the students says, "Don't panic, with a little discipline we will have a $\frac{9}{10}$ chance of going on vacation before the end of the week." Can you find his reasoning?
This problem comes after the chapter permutation group $S_n$. My thought is in previous turns we may have one or more members leave and go for vacation provide they are lucky. And as they try more and more times the probability of finding their names increases since more and more members are left. But to get $\frac{9}{10}$ is a really hard job for me.
The original question is ambiguous at some point. For example, if the headmaster call the first member into the amphitheater 250 times consecutively, then the first member can find it's name with chance $\frac{1}{2}$, so it would neve reach $\frac{9}{10}$. So the only way which make this possible is the headmaster call the 500 members in turns: each turns the members from 1 to 500 will go to the amphitheater one by one. So the question want you to show that they can all go for vacation before 250 turns over. Another thing remarkable is once a person find his name then he will leave immediately and the others can see that who is left.
this might be related to the locker puzzle. the probability of everybody going on vacation on the $k^{th}$ day is about $0.3*0.7^{k-1}$. summing over k=1 to 7, we have the probability 0.9176.