Maximum likelihood estimation problem

Question

In a town, there are taxis labeled from $1$ to $n$ ($n$ is the total number of taxis). We arrive in the town and see a taxi with label $100$. With the method of maximum likelihood estimation, find $n$.

The solution says $n=100$, but I can't figure out why.

Answer 1

The likelihood on this data is $1/n$ if $n\ge 100$ and zero otherwise (since a draw of $100$ is impossible if $n<100$). This is maximized for $n=100,$ so the MLE is $100.$

Answer 2

The probability that the number of the first taxi observed is $100$ is $$ \begin{cases} 1/n & \text{if there are $n$ taxis and $n\ge100$}, \\ 0 & \text{if there are $n$ taxis and } n<100. \end{cases} $$ So the probability is highest when $n=100,$ so that is the maximum-likelihood estimate.