I came across the following problem (Problem number 27 from here): Aaron samples from the Uniform(0,1) distribution. Then Brooke repeatedly samples from the same distribution until she obtains a number higher than Aaron’s. How many samples is she expected to make?
Here's my attempt: Let the number of draws Brooke makes be $N$. Let Aaron draw $A$. Then given $A=a$, $N$ is a Geometric($1-a$) random variable, with expectation $1/(1-a)$. Hence,
$$ E[N] = E[E[N|A]] = E\left[\frac{1}{1-A}\right] $$
Here's where I got stuck, because I don't think for $A\sim U(0,1)$, $E[(1-A)^{-1}]$ is finite.
You can check the link for a solution provided by the person who wrote the article (their answer is $\pi^2/6)$, but I don't see the flaw in my logic. Is the answer really $\infty$?
Related, but not exactly same problem: Ms. A selects a number $X$ randomly from the uniform distribution on $[0, 1]$. Then Mr. B repeatedly, and independently, draws numbers