A football team enters a tournament in which they play one game with each of three teams, but they get to choose the order in which they play their opponent teams. The team knows the probability of "win" against each of the three teams:
PA: is the probability to win when playing with team A
PB: is the probability to win when playing with team B
PC: is the probability to win when playing with team C
where pC
The team wins the tournament if they win 2 games in a row. The team wants to maximise the probability of winning. Show that the "win" probability is maximised if they play the weakest team second, and that the order of playing the other two teams doesn’t matter.