You pull cards out of a shuffled deck until you get a jack and a queen. Given see the jack first, what is the expected number of cards between the jack and the queen?
I believe this is a conditional expectation problem. Our goal is to subtract the expected position of the jack from the expected position of the queen. However, I am lost on how to find this other than writing out all possible combinations which isn't feasible. I know how to find the expected position of either card without the condition. Do I use this and compute variance in some way? Kind of lost, any guidance is appreciated.
Best Answer
Suppose that the first J occurs in position $i\in[45]$. Then, there are $52-i$ cards remaining, $4$ of which are queens. A classical result shows that the expected number of cards to flip the first queen must then be $(48-i)/5$. Now the probability that the first J occurs in position $i\in[45]$ is given by $\binom{52-i}{7}/\binom{52}{8}$, so the expectation is $$ \sum_{i=1}^{45}\frac{48-i}{5}\frac{\binom{52-i}{7}}{\binom{52}{8}} = \frac{379}{45} \approx 8.42 $$ (evaluated using WolframAlpha).