Find the Conditional Distribution of X

Question

Let N have a Bin(100, 1/4) distribution. Given that N = n, flip a fair coin n times, and let X be the number of heads observed. Find the conditional distribution of X given that N = n. Be precise about the possible values and a proper conditional probability mass function

I'm really stumped on this one. So far I concluded P(X | N=n) = P(X, N=n)/P(N=n). I don't know where to go from here. I also know that the p.m.f./p.d.f. of Bin(n,p) is (nCk)(p^k)(1-p)^(n-k) but I don't know how to apply it to this question.

Answer 1

This is simple and you're overthinking it. Conditional on $N=n$ just means that we are considering the case when there are $n$ flips of the fair coin. The conditional distribution is thus $Bin(n,p)$ so $$ P(X=k \mid N=n) = \frac{1}{2^n}{n \choose k}$$ for $k=0,1,\ldots n.$

Answer 2

Don't over think.

When given that $n$ coins have been fliped, what is the distribution for the count of heads among those $n$ flips?