Take a random walk

Imagine you are sitting in a circle of n + 1 friends, and positions are labelled 0, 1, … n, and so on. The B in Fig 1 represents a pizza box held by person 0, who can pass the pizza to the person on their left or right with equal probability. That person randomly passes the pizza to their left or right, and so on. After a given number of passes, everyone in an arc of the circle has touched the pizza, and everyone outside the arc has not. The arc grows until all but one person has touched the pizza – and they are the winner. Where in the circle should you position yourself to maximise the probability that you win?

– Courtney Linder