There's a game with four players, numbered by their increasing difficulty:
The probability of any player (
x) beating another player (
y) in a 1v1 game is
x / (x + y). For example if player
3 plays player
4 the chance of player
3 winning is
3/7. There are no draws - every game ends in one side winning.
There's a tournament where in the first round the four players are organised in to two 1v1 groups (group
A and group
B). The winner of each group play each other in the final. The winner of the final wins the tournament.
1 gets to choose who they want to play in the first round. For example they might choose to play the easiest opponent (
2), and hope that they don't meet the hardest opponent (
4) in the final.
The possible groupings of the first round are:
A: 1v2 B: 3v4
A: 1v3 B: 2v4
A: 1v4 B: 2v3
And in the final the winner of group
A will play the winner or group
1 increase their chance of winning the tournament through their choice of opponent in the first round? (And how can I calculate their chances in each scenario?)