What is the probability that given a bin (out of $n$ bins) that it will receive no balls I'm working on a problem where there are $n$ bins. I want to know the probability that given a bin (Specific bin) that it will receive zero balls. There are $m$ balls thrown at random into the bins.
This is similar to my homework problem, but I'm looking for the formula. I'll input my specific variables after I understand it. thanks

The answer depends on whether the balls are distinguishable.


*

*Suppose that the balls are indistinguishable. Using stars and bars, the number of ways we can put $m$ balls into $n$ bins is equal to
$$
{n+m-1\choose n-1}.
$$ 
The number of ways to put $m$ balls into $n-1$ bins (we select a specific bin and do not put any balls into it) is equal to
$$
{n+m-2\choose n-2}.
$$
The probability that the selected bin will contain $0$ balls is then given by
$$
\frac{n+m-2\choose n-2}{n+m-1\choose n-1}=\frac{n-1}{n+m-1}.
$$

*Suppose that the balls are distinguishable. Then the number of ways we can put $m$ balls into $n$ bins is equal to
$$
n^m.
$$
The number of ways to put $m$ balls into $n-1$ bins is equal to
$$
(n-1)^m.
$$
The probability that the selected bin will contain $0$ balls is then given by
$$
\biggl(\frac{n-1}n\biggr)^m.
$$
