Solution: 2017 Winter Midterm - 17

Author: Michiel Smid

Question

Let $n \geq 1$ be an integer. A bag contains $n$ red balls and $n$ blue balls. We choose a uniformly random subset of two balls. Define the event

What is $\Pr(A)$?

(a)

$\left. n^2 \middle/ {n \choose 2} \right.$

(b)

$\left. {2n \choose 2} \middle/ n^2 \right.$

(c)

$\left. n^2 \middle/ {2n \choose 2} \right.$

(d)

$\left. {n \choose 2} \middle/ n^2 \right.$

First, we have n red balls to choose from and n blue balls to choose from.

Thus, there are $ n \cdot n $ ways to choose a subset of two balls.

There are a total of $ \binom{2n}{2} $ ways to choose a subset of two balls.

$ Pr(A) = \frac{n \cdot n}{\binom{2n}{2}} $

$ Pr(A) = \frac{n^2}{\binom{2n}{2}} $