Probability of throwing balls into bins

Question

You are throwing n balls into m bins randomly.

1. What is the probability to be empty of the first $k$ bin?

2. Given $k$ bins are empty. What is the probability to be empty of $(k+1)th$ bin?

3. Forget the first 2 cases, let throw balls into $m - k$ bins. What is the probability to be empty of the first bin?

Thank you

Answer 1

It is helpful to imagine the balls are distinguishable. That makes no difference to the probability.

There are $m^n$ equally likely ways to distribute the $n$ balls among the $n$ bins. There are $(m-1)^n$ ways to distribute the balls among the last $m-1$ bins. Thus the probability the first bin is empty is $\frac{(m-1)^n}{m^n}$.

Or else when we throw a ball, the probability it misses Bin 1 is $\frac{m-1}{m}$. The probability of missing Bin 1 $n$ times in a row is $\left(\frac{m-1}{m}\right)^n$.

The other problems are solved using similar reasoning.