Question

Question: When \(m\) balls are distributed into \(n\) bins uniformly at random, what is the probability that the first bin remains empty?

Short answer

Answer

The resultant answer is\({\left( {\frac{{n - 1}}{n}} \right)^m}\).

Step-by-step solution

Given data 

The given data is \(m\) balls and \(n\)bins.

Simplify the expression

When \(m\) balls are distributed into \(n\) bins uniformly at random.

Probability that the first ball is not placed in the first bin \( = \frac{{n - 1}}{n}\).

The probability that a ball is not placed in the first bin is same for each of the \(m\) balls, then the probability that the first bin remains empty \( = {\left( {\frac{{n - 1}}{n}} \right)^m}\)