An insect lays eggs. The number of eggs laid is a random variable $X$ that follows a Poisson distribution of intensity $θ$. Each egg has a probability $p$ of hatching, independent of other eggs. Let $Z$ be the random variable that determines the number of eggs that hatched.
a) For $(k, n) ∈ \mathbb{N}^2$, calculate $P [Z = k | X = n]$.
b) Using the total probability formula, find $Z$'s law.
Hi, I've stuck in the part $b)$, I know what the formula means, but can't seem to find a way to use it for finding $Z$'s law
Can someone explain, how to use the formula in this case?
I think you just want $$P(Z=k) = \sum\limits_n P(Z=k|X=n)\cdot P(X=n)$$