Picking Multiples of 4 I recently came up with and tried to solve the following problem: If you are randomly picking integers in the range $[1,30]$ out of a hat without replacement, on average, how many integers will you have to pick until you have picked all of the multiples of $4$?
There are $7$ multiples of $4$ that can be chosen. I know that the expected value of picks until you pick a multiple of 4 is the smallest value of $n$ such that $1-\displaystyle\prod_{i=0}^{n-1}\dfrac{23-i}{30-i}>0.5$, which is $3$. However, I don't know how to figure out how many picks are needed until all multiples of $4$ have been chosen. Can I please have some assistance?

Another approach would be to think of a permutation of $\{1, 2, ..., 30\}$.  We can think of the seven multiples of $4$ as separating runs (some possibly empty) of the $23$ nonmultiples of $4$.  Since there are seven separators, there are $8$ runs, with the average run length being $\frac{23}{8}$.  Thus the average number of nonmultiples of $4$ after the last (seventh) multiple of $4$ is $\frac{23}{8}=2.875$.  Thus the average position of the last multiple of $4$ is position $30-2.875=27.125$.  
