TITLE: What is the expected number of red apples left when all the green apples are picked?
QUESTION [6 upvotes]: We have 4 green apples and 60 red apples. Each time we pick one out without replacement. Then what is the expected number of red apples left when all 4 green apples are picked?

REPLY [2 votes]: The probability that $k$ or more red apples remain is
$$
\frac{60}{64}\frac{59}{63}\cdots\frac{61-k}{65-k}=\frac{\binom{64-k}{4}}{\binom{64}{4}}
$$
So the probability that exactly $k$ red apples remain is
$$
\begin{align}
\frac{\binom{64-k}{4}-\binom{63-k}{4}}{\binom{64}{4}}
\end{align}
$$
and the expected number of red apples would be
$$
\begin{align}
&\frac1{\binom{64}{4}}\sum_{k=1}^{60}k\left(\binom{64-k}{4}-\binom{63-k}{4}\right)\\
&=\frac1{\binom{64}{4}}\left[\sum_{k=0}^{59}(k+1)\binom{63-k}{4}-\sum_{k=1}^{60}k\binom{63-k}{4}\right]\\
&=\frac1{\binom{64}{4}}\left[\sum_{k=0}^{59}(k+1)\binom{63-k}{4}-\sum_{k=0}^{59}k\binom{63-k}{4}\right]\\
&=\frac1{\binom{64}{4}}\sum_{k=0}^{59}\binom{63-k}{4}\\
&=\frac1{\binom{64}{4}}\binom{64}{5}\\[6pt]
&=12
\end{align}
$$