Balls and Bins problem

Question

Below is the problem I wanted to solve :

If m balls are thrown randomly into n bins, what is the probability that at 
least two balls end up in the same bin? (n < m)

Is it the same problem as Balls and Bins ?

My instructor tells me the solution is 1 - (nCm * m!)/n^m. I am little confused.

It is not the same problem. That one wants the number of bins with exactly one ball; you want no bins to have two or more balls. — Ross Millikan, Oct 23 '13 at 20:30
@RossMillikan It asks for strictly more than one ball, which I figure is same as at least two balls? — user2761431, Oct 23 '13 at 20:32
But it wants the expected number, while you want the chance of none — Ross Millikan, Oct 23 '13 at 20:36
Yeah, I just want the probability and I want to know if @Henry probability value to that question is same. — user2761431, Oct 23 '13 at 20:39
If there are more balls than bins then the answer is $1$. Otherwise https://en.wikipedia.org/wiki/Birthday_problem#Calculating_the_probability shows how to do the calculation — Henry, Oct 23 '13 at 20:42

score 0 · Answer 1 · answered Oct 23 '13 at 23:52

Consider the complement. If no set of two balls end up in the same bin, it means that each ball is an individual bin.

How many ways are there to distribute $n$ balls into $m$ bins, with at most 1 ball in each bin?

How many ways are there to distribute $n$ balls into $m$ bins, without restriction?

Hence, conclude that your instructor is correct.

1 Answers1