Question: 2018 Winter Final - 20

Author: Michiel Smid

Let $n \geq 2$ be an integer. You are given $n$ beer bottles $B_1,B_2,\dots,B_n$ and one cider bottle $C$. Consider a uniformly random permutation of these $n+1$ bottles. The positions in this permutation are numbered as $1,2,3,\dots,n+1$. Define the random variable $X$ to be

X = the position of the leftmost beer bottle.

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

(a)

$\frac{n+3}{n+1}$

(b)

$\frac{n+1}{n}$

(c)

(d)

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