A box contains $2n$ balls of $n$ different colors, with 2 of each color. Balls are picked at random from the box with replacement until two balls of the same color have appeared. Let $X$ be the number of draws made.
a) Find a formula for $P(X>k)$ $k=2,3,...$
b) Assuming $n$ is large, use an exponential approximation to find a formula for $k$ in terms of $n$ such that $P(X>k)$ is approximately 1/2. Evaluate $k$ for n equal to one million.
My thought: For part a)
$P(X>k) = 1-P(X\le k) = 1-P(X=0)-P(X=1)-P(X=2) = 1-0-0-\dfrac{1}{2n-1} = 1-\dfrac{1}{2n-1} $
I get stuck on the part b because of the answer for part a.
Could someone help me out?