Intuition behind independence result The following problem is from Wasserman's $\textit{All Of Statistic}s$. I have worked through the algebra to arrive at the result, but it still seems very strange to me, so I would appreciate any intuition that could be offered.
Let $N$~Poisson($\lambda$). Suppose we toss a coin $N$ times, with $p$ being the probability of heads, and let $X$ be the number of heads, and $Y$ be the number of tails. Show that $X$ and $Y$ are independent.
Now, as I said, I have shown that $X$ and $Y$ are independent, but it $\textit{feels}$ like they shouldn't be. At first glance it seems like a very large value of $X$ should make $P(Y|X)$ smaller for large $Y$. What's going on here?

Suppose you're talking about a fair coin, and you're told that 50 tosses (from an unknown number of tosses) are heads. Intuitively, to infer $Y=N-X$ given $X=50$, say, you would need to first infer $N$, which in turn is inferred based on the probability of heads relative to tails (the next best thing you have as you don't know X), and this probability has absolutely nothing to do with your observed values.
Essentially, the detail is that you have no idea about what the entire sample size is, so if you are only given that a million people in city X are male, the best guess you have for the number of females is also a million, i.e. $\mathbb P [\text{individual is female}]=0.5$ regardless of the number of males in the city when you don't know the full population.
