I'm confused by the reasoning for the answer to subproblem (c) from Blitzstein and Hwang,chapter 3, problem 27:
A company with n women and m men as employees is deciding which employees to promote.
(a) Suppose for this part that the company decides to promote t employees, where 1 <= t <= n + m, by choosing t random employees (with equal probabilities for each set of t employees). What is the distribution of the number of women who get promoted?
(b) Now suppose that instead of having a predetermined number of promotions to give, the company decides independently for each employee, promoting the employee with probability p. Find the distributions of the number of women who are promoted, the number of women who are not promoted, and the number of employees who are promoted.
(c) In the set-up from (b), find the conditional distribution of the number of women who are promoted, given that exactly t employees are promoted.
Now for (c), assuming that Y is the number of promoted women; X is the number of promoted; t is the total number of promoted employees, I have that:
$$ P(Y=k|X+Y=t) = \frac{P(Y=k \cap X+Y=t)}{P(X+y=t)} = \frac{P(Y=k)}{P(X+Y=t)} = \frac{\binom{n}{k}p^k(1-p)^{n-k}\binom{m}{t-k}p^{t-k}(1-p)^{m-t+k}}{\binom{n+m}{t}p^t(1-p)^{n+m-t}} $$
Which simplifies to the hypergeometric: $\frac{\binom{n}{k}\binom{m}{t-k}}{\binom{n+m}{t}}$
What I can't wrap my head around is why the numerator of the conditional probability, $P(Y=k \cap X+Y=t)$ simplifies to $P(Y=k)$ and why that is the product of the $Bin(m, p)$ and $Bin(n, p)$, instead of just $Bin(m, p)$.
First, for the simplification, I suspect it has something to with $P(X+Y=t)$ begin given, so for the intersection of two events, A and B, if B is given and we're conditioning on B, then that would simplify $P(A \cap B)$ to $P(A)$?
And second, for why the male binomial is introduced, I think what's confusing me is that in subproblem (b), we are asked to Find the distribution of women that are promoted, so I make that into r.v. $Y \sim Bin(m, p)$. So then in (c), I would expect $P(Y=k)$ to simply be the binomial of $Y \sim Bin(m, p)$, but instead it also includes the male binomial. Is it because we're assuming X+Y=t, so that we're looking for Y female promotions and X male promotions? (Ie. am I confusing dependence/independence here?)
Is there a simpler way, a technique, to conceptualize this more intuitively?
I looked at two other answers to understand, PMF of HGeom & Binom Distributions (Women and Men getting promoted) and Determining the distribution of women in a random sample of unknown size from a population size of n women and m men but have had no luck, except in verifying the math behind the problem.