Poisson and conditional distribution

Question

Given a source that emits photons let be $X$ the number of photons emitted in $s$ seconds. Let be $p$ the probability that photons reach a detector and let be $Y$ the number of photos that reached the detector after $s$ seconds.

I need to find:

1. Distribution of $X$;
2. $\mathbb{P}(Y=y \mid X=x)$;
3. Marginal distribution of $Y$.

Q1

$X\sim\mathrm{Po}(\lambda s)$ $$\mathbb{P}(X=x)=e^{\lambda s}\dfrac{(\lambda s)^x}{x!}$$

Q2

Not sure about this:

$$\mathbb{P}(Y =y\mid X=x)=\binom{x}{y}p^y(1-p)^{x-y}$$

Q3

Any suggest?