Soccer and general probability distribution? Three teams (Team 1, Team 2, and Team 3) are playing in a soccer championship. In each game, two teams play each other and the third team does not play. The winner of any given game $n$ plays again in game $n+1$ against the team that did not play in game $n$, and the loser of game $n$ does not play in game $n+1$. The probability that Team 1 will beat Team 2 in any game they play against each other is $0.3$. The probability that Team 1 will beat Team 3 in any game they play against each other is $0.6$. The probability that Team 2 will beat Team 3 in any game they play against each other is $0.8$. (Assume that game always results in
either win or lose)
(a) Determine the probability that the two teams who play against each other in the
first game will play against each other in the fourth game.
(b) Show that this probability does not depend on which two teams play in the first
game.

Hint: there are three states of game play: 1 vs. 2, 1 vs. 3, and 2 vs 3.  For each state, you know the probability of which state the next game will be.  Given that, what would the states of games have to be to have the fourth game be the same state as the first game?
