Question

1) What is the probability of having every bin filled with at least one ball if n balls are distributed randomly in to m bins ? Do all case m < n, m = n , m > n

Answer 1

Total number of ways to distribute n balls into m bins is computed here as:

= m*m*m..... n times
= mⁿ

a) For m < n,

The number of ways to distributed n balls into m bins is computed using the multinomial formula which is equivalent to the number of ways to divide n items into m groups such that each group has at least 1 item given that all items are identical. It is computed here as:

But as the balls are not identical, we also need to multiply it with the permutation of number of balls that is n! here.

Therefore the probability here is computed as:

b) For m = n, number of ways to place n balls into n bins is computed as the number of permutation of n that is n!

Therefore the probability here is computed as:

This is the required probability here.

c) For m > n, we can never have at least one ball in each of the m bins, therefore the probability here is computed as 0.