Suppose we draw cards from a standard deck one-by-one. Find the
expected number of cards after the first Jack and before the first
Queen given that the first Jack appears before the first Queen.
My approach is that we label all 4 Jacks and 4 Queens in the deck as special cards. Now there are 44 other cards in the deck to be inserted in 9 places. Therefore on average there are $\frac{44}{9}$ cards between two consecutive special cards.
Given that first Jack comes before first Queen, there are $\frac{8!}{2}$ ways that can happen in special cards. Now, let $P_k$ be the probability that first Queen is the kth card after first Jack in our special cards. Therefore the total expected number of cards between first Jack and first Queen is $$\frac{44}{9}(P_1 \cdot 1 + P_2 \cdot 2 + P_3 \cdot 3 + P_4 \cdot 4) = \frac{352}{45}$$ but I know that the actual answer is $\frac{379}{45}$. What is the problem in my approach?
Best Answer
The problem is that you have counted the expected length of gaps between Jacks and Queens or between Jacks and Jacks with $\frac{52-8}{9}$ for each such gap, but have not counted those Jacks which come after the first Jack and before the first Queen. Try $$P_1\cdot\left(0+\frac{44}9 \cdot 1\right) + P_2\cdot\left(1+\frac{44}9 \cdot 2\right) + P_3\cdot\left(2+\frac{44}9 \cdot 3\right) + P_4\cdot\left(3+\frac{44}9 \cdot 4\right)$$ to get $\frac{379}{45}$.