TITLE: German Tank Problem, Confidence Level
QUESTION [6 upvotes]: Suppose you're in a city with n cabs. Each cab has a distinctive number from 0 to n. You take a cab 10 times, the choice of the cabs is independent and equiprobable. The cab with the biggest number has number 100.
Determine a preferably small upper limit for n with a confidence level of $\frac{9}{10}$.
To me, this sounds like the German Tank Problem. So, depending on the approach I get different estimates for n. But I don't know how to include the confidence level. Do you have any ideas?

REPLY [6 votes]: If they are numbered $0$ to $n$, that's $n+1$ numbers so there's one missing.  To make the problem nicer, I'll suppose they are numbered $1$ to $n$.
A statistic $L$ is an upper limit for $n$ with confidence level $p$ if for every possible $n$, $P(L \ge n) \ge p$.
In this case if $M$ is the maximum number observed and $L = c M$ where $c > 1$ is a constant, 
$$P(L \ge n) = P(M \ge n/c) = 1 - \left(\dfrac{\lceil n/c \rceil - 1}{n}\right)^{10} \ge 1 - \dfrac{1}{c^{10}}$$
To make the right side $9/10$, we want to take $c = 10^{1/10}$.  Thus for the observed $M = 100$
we would have $L \approx  125.89$, which could be rounded down to $125$.