Expected number of collisions
I posted this thread in the 'pre-university probability and statistics'-section and so far there have not been any replies. Probably I should have posted it it in the advanced section in the first place.
I have had a hard time trying to solve the following problem. I would be pleased if anyone can help me on this (this problem's description actually originates from a hashing problem, known in computer science, which I have tried to simplify by describing a more general understandable situation):
There are m open boxes, each of which can contain at most one ball. There is a ball throwing machine that fires a total of n balls towards the boxes, one by one. For each box, the probability that when the machine throws a ball, the ball falls onto that specific box, is 1/m (equal for each box). The thrown ball will always fall onto one of the boxes. However, the probability that the ball will actually fall into any box is less than 1 after the first throw (provided that n > 1), because a box which is already filled with a ball (from the previous throw(s)), will prevent the thrown ball from falling into that box. Call this a collision. Assume that a thrown ball, which collides with one of the balls in a box, simply disappears.
I need a formula with which I can calculate the expected number of collisions, given m boxes and n balls.
Thanks in advance.