A number of people (N) are visiting a hotel. They each get a room with a key.
They go to dinner and have to put their keys in a big container. When they return from dinner (possibly drunk) they each reach in and pick a key at random.
What is the probability that at least one will go back to his original room?

What happens when the number of people is very large?

N.

How different is this from the birthday problem?

Re read it.
Here each guy gets a complete different room, two people cant go back to the same room. It is completely unrelated.

When the number of people is very large, the number of rooms will be very large, and hence the number of keys will be very large, and hence the container will need to be very large.

What happens is that:

a) it takes a very long time to put the keys into the container (maybe more than 1 week)
b) the container, when full, is very heavy (and would probably need to rest on a sturdy table rather than being held in the hands)
c) it takes a long time to take the keys out of the container (cf: a) above)

All in the container.

First one takes a key. Either theirs (1/n) or not n-1/n)
say not...

Next one take a key. Either theirs (1/n-1) or not (n-2/n-1)

seems pretty darn likely that someone will get their key, as for this NOT to happen would require each get a wrong key and that gets less likely with each pick.

Jonathan,

It's not that easy since your second probability (second person taking their key) is dependant on if the first person picked the second person's key or not. If the first person picks the second person's key the conditional probability of the second person taking his own key is 0. This is the same as the Hat Problem and the math behind the classic card trick Frequent Miracle which was discussed at

http://www.themagiccafe.com/forums/viewt......forum=99

/Tomas

Tomas - by Hat Problem, do you mean me?

Aren't the odds ((number of people) * (number of keys))=chance of getting your key?

Hint - it has something to do with "e" ...

E by gum?

Stan, Tomas ?
This one's for you guys.
This is a very beautiful problem with a somewhat strange outcome...
nir

Tomas already answered this, didn't he?
(Or at least provided a link where the answer can be found.)

Dang,

Didn't see that - that is correct.
Guess I'll have to find another goodie for you guys.