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**Statistics problem---exponential approximation**

## Homework Statement

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.

## Homework Equations

## The Attempt at a Solution

For part a), I got that P(X>k) = (2n-2)/2n * (2n-4)/2n *...* (2n-2k+2n)/2n for k terms.

For part b), how do I set up an exponential approximation? To get started, I think that it would be e^(-1) + e^(-2) +...+ e^(1-k)... am I on the right track?