Here’s a problem for the more math-inclined among you.
There’s a party downtown, and all the city’s richest and snobbiest people are there. Being fashionably minded, they all have unique hats, but since they are all lazy and don’t want to carry them around, they leave them at the door with the attendant. A certain number of people show up at this party, let’s say a nice round number like n. But being rich and snobby, none of them leave a tip with the attendant.
So the party starts to come to a close, and all the guests start to leave. Each one goes to pick up his or her hat on the way out. But since they were all such lousy tippers, the attendant, in a fit of pique, decides to hand each guest a hat at random and shove that guest out the door.
The question is this: on average, how many people from the party got their own hats back? (In other words, if you were to do this experiment a hundred times, or a thousand, what’s the average number of people who would get their own hats back?)
Hints to come later, solutions after that…
As a non-math person I have two answers:
1. They were all drunk and didn’t care what hats they got.
2. They thought it was the new “in” game to trade hats, and this started a trend.
3. They might think it’s a new game of hat roulette and the one who gets his own hat is forever banned from the group.
Perhaps he is forced to eat his own hat.
well, after running about 10000 sims for a 20-person party, i found that one person got their own hat on average. i could maybe actually figure out a general solution, but i’d first be interested to see if that answer is correct.
and now, i have to go pay those 20 people for going to 10000 parties with pissed-off hat-checkers
Yes, that is correct. Interestingly enough, it does not matter how many people are at the party; on average exactly one person will get their hat back.
Even more interesting is the fact that regardless of what order they go in, each person has an identical chance of getting his or her hat back: 1 in n.
That is quite an interesting problem. Wherever did you hear of it? Surely it sprung from a true fountain of knowledge.
Hehe, yep, this is really Andrew’s idea… I figured I’d see if any non-mathy folks could figure it out.