Empty Pigeonholes

Suppose we have \(N\) pigeons and \(N\) holes, and the pigeons are uniformly distributed amongst the various holes. What is the expected number of empty holes?

Solution

Define \(X = \text{number of empty holes} = \sum_{n=1}^N \mathbb{I}\{\text{nth hole is empty} \}\)

Then

\[\begin{align*} \mathbb{E}[X] &= \sum_n \mathbb{P}(\text{nth hole is empty})\\ &= \sum_n \Big(\frac{N -1}{N} \Big)^N\\ &= N(1 - \frac{1}{N})^N\\ &= N / e \end{align*}\]

because \(1 - \frac{1}{N} \approx \exp(-1/N)\).