Tom is searching for the 6 books he needs in a random pile of 30 books. What is the expected number of books must he examine before finding all 6 books he needs (PUMAC Combo Round 2007)?
I'm just wondering if I have the correct approach to this problem. This feels really similar to that one question asking the expected number of cards you draw before you see an Ace so I took a similar approach. Let $X_0, \ldots, X_6$ be the number of books drawn in between the 6 books of interest. Basically what I'm saying is that we draw $X_0$ books and then draw the first book of interest. Then we draw $X_1$ books and then the second book of interest, and so on and so forth.
The answer to the question should be $6 + \mathbb{E}[X_0+X_1+X_2+X_3+X_4+X_5]$. All of these are the same by symmetry and $\mathbb{E}[X_0] = (30-6)/7 = 24/7$. Thus, the answer is $6\cdot 24/7 + 6 = \boxed{186/7}$.