Let $k$ be a fixed positive number. We toss a normal coin until we get at least $k$ heads and at least $k$ tails (not necessarily consecutively). Let $X$ be number of needed tosses. Find distribution of $X$ and its expected value.
So this is my attempt: first of all, we see that $P(X=j)$ is $0$ for $j<2k$. For $j \ge 2k$ we have the following:
Let A be the event that in the last toss we get a head.
Let B be the event that in the last toss we get a tail. So:
$P(X=j) = P(X=j | A) P(A) + P(X=j|B) P(B) = $ ${j-1}\choose{k-1}$$ (\frac{1}{2})^{j-1} \frac{1}{2}$ + ${j-1}\choose{k}$$ (\frac{1}{2})^{j-1} \frac{1}{2} = (\frac{1}{2})^j [$$ {j-1}\choose{k-1}$ ${j-1}\choose{k} $$] = (\frac{1}{2})^j $${j}\choose{k}$
So I found the distribution of $X$. But the problem comes with the expected value:
$E(X) = \sum_{j=2k}^\infty j (\frac{1}{2})^j $${j}\choose{k} $=...
How to compute it? May you help me and show me how?