medium Probability

Ball Out

A release-checked medium problem for training Law of Total Probability, Markov Chains, Events.

Question

You and $4$ other people are standing in a pentagonal shape with you at the top vertex. You are given a basketball to start the game. Each minute, the person with the ball has three options: pass left, pass right, or keep. Each choice is selected with a uniformly random probability, independent of previous decisions. This repeats until someone keeps the ball. What is the probability that you are the person to keep the ball?

Practice focus

This Probability problem is tagged Law of Total Probability, Markov Chains, Events. State the random variables and conditioning information explicitly, then check the result against boundary cases before opening hints or a solution.