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?

The probability of getting all multiples of $4$ precisely on draw $k$ is
$$
\frac{\binom{k-1}{6}}{\binom{30}{7}}
$$
Thus, the expected number of draws would be
$$
\begin{align}
\frac1{\binom{30}{7}}\sum_{k=7}^{30}k\binom{k-1}{6}
&=\frac7{\binom{30}{7}}\sum_{k=7}^{30}\binom{k}{7}\\
&=\frac7{\binom{30}{7}}\binom{31}{8}\\
&=\frac{217}{8}\\[9pt]
&=27.125
\end{align}
$$
