# Conditional binomial distribution

There are $$n$$ eggs, each of which hatches a chick with probability $$p$$ (independently). Each of these chicks survives with probability $$r$$, independently. What is the distribution of the number of chicks that hatch? What is the distribution of the number of chicks that survive? (Give the PMFs; also give the names of the distributions and their parameters, if applicable.)

I'm trying to answer this as mathematically rigorously as possible.

Let $$X$$ be the number of chicks that hatch and let $$Y$$ be the number of chicks that survive. It's obvious that $$X \sim Bin(n, p)$$, so the PMF of $$X$$ is: $$p_X(x) = P(X=x) = {n\choose x}p^x(1-p)^{n-x}$$ I tried to define and calculate the conditional distribution of $$X\,|\,Y$$, where the number of chicks that survive is conditional on the number of chicks that hatch. $$p_{Y|X}(y|x) = P(Y=y|X=x) = \frac{P(Y=y, X=x)}{P(X=x)}$$.

I couldn't figure out how to express $$P(Y=y,X=x)$$. $$P(Y=y)P(X=x|Y=y)$$ didn't seem to make much sense.

$$X$$ is $$\text{Bin}(n,p)$$, not $$r$$. And, given number of chicks ($$X=x$$), the number of surviving chicks is already a Bernoulli with $$n=x$$ and $$p=r$$. You don't need to use conditional probability formula for that. But, the question is to find "distribution of number of chicks that survive", i.e. $$P_Y(y)$$, instead of $$P_{Y|X}(y)$$.
An easy way to think is each egg has $$pr$$ probability of becoming a surviving chicken, let's call it as $$Y_i$$, and $$Y=\sum Y_i$$, therefore again Binomial distributed, i.e. $$\text{Bin}(n, pr)$$.
• Well, since the data generation process goes from $X$ to $Y$, the easiest thing to come up is $p(y|x)$. You can write $p(y,x)$ but the straightforward thinking leads from $p(y|x)p(x)$. Nov 2, 2020 at 13:10