Question:
There are $n$ flower buds in a garden, each of which opens (independently) with a probability $p$ and each of the blooms survive (independently) with a probability $r$. What's the distribution of the number of buds that survive as a flower?
Attempt:
Let $X$ be the number of buds that open and $Y$ be the number of flowers that survive. Then $X \sim \text{Bin}(n,p)$ and $Y|X =k \sim \text{Bin}(k,r)$. I think eventually we will get $Y \sim \text{Bin}(n,pr)$, but I don't know how to get there.
$$P(Y = y) = \sum_{k=y}^{n} P(Y = y | X = k)P(X=k) = \sum_{k=y}^{n} {k \choose y}r^y(1-r)^{k-y}{n \choose k}p^k (1-p)^{n-k} = ?$$
