SIMON-MARAIS 2018 A4

For each positive integer $n$, consider a cinema with $n$ seats in a row, numbered left to right from 1 up to $n$. There is a cup holder between any two adjacent seats and there is a cup holder on the right of seat $n$. So seat 1 is next to one cup holder, while every other seat is next to two cup holders. There are $n$ people, each holding a drink, waiting in a line to sit down. In turn, each person chooses an available seat uniformly at random and carries out the following.

• If they sit next to two empty cup holders, then they place their drink in the left cup holder with probability $\frac{1}{2}$ or in the right cup holder with probability $\frac{1}{2}$.
• If they sit next to one empty cup holder, then they place their drink in that empty cup holder.
• If they sit next to zero empty cup holders, then they hold their drink in their hands.

Let $p_n$ be the probability that all $n$ people place their drink in a cup holder.

Determine $p_1 + p_2 + p_3 + \cdots$.