Question: 2019 Winter Final - 21

Author: Michiel Smid

Let $n \geq 2$ be an integer. You are given $n$ beer bottles $B_1,B_2,...,B_n$ and two cider bottles $C_1$ and $C_2$. Consider a uniformly random permutation of these $n + 2$ bottles. The positions in this permutation are numbered $1,2,...,n + 2$. Consider the random variable

X = the position of the leftmost beer bottle.

What is the expected value $\mathbb{E}(X)$ of the random variable $X$?

(a)

$\frac{n}{n + 2} + \frac{2n + 6}{(n + 1)(n + 2)}$

(b)

$\frac{n}{n + 2} + \frac{4n + 2}{(n + 1)(n + 2)}$

(c)

$\frac{n}{n + 2} + \frac{4n + 6}{(n + 1)(n + 2)}$

(d)

$\frac{n}{n + 2} + \frac{2n + 2}{(n + 1)(n + 2)}$