You are one of ten friends who are going to play a few rounds of a simple two-player game of chance. Each player has a 50% chance of winning the game each round.
There are 5 playing squares on the floor, arranged in a circle.
In the beginning, two people go to each square. Each round the people at a square play the game and determine the winner. A round ends when the winner walks clockwise to the next square. The loser stays where they are.
After 10 rounds, what is the probability you are on the square you started on?
Two expert and jaded tic-tac-toe players, after drawing for the n
-th time, decided to add some randomness to their favorite game.
First, they used a coin to decide who would start. Then, that player would pick his initial move randomly. Next, the other player would also pick his answer randomly. Finally, from then on the game went on as usual, with each player playing in the best possible way.
For each player, what are the odds of winning, losing, or drawing?