A Fair Payout
I organize a game between 2 equally talented players, Arya and Bran, consisting of 5 rounds. The first person to win 3 rounds will win $120.
Suppose that the game was interrupted at the end of the third round, with Arya leading 2-1. We would like to split the prize in a fair way between these two players. How much should Arya get?
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1 solution
I think the idea here is that both Arya and Bran have equal chance ( 50-50) of winning each of the 3 rounds. Since Arya is leading 2-1, the initial thought was to divide the $120 into 3 parts with Arya keeping 2/3 of it or $80. But if you take a step back and think through this, Bran has still got some chance to win the entire game, provided he wins the 4th and 5th round. If Bran would to win it all, he has to win the next 2 consecutive games, or probability wise, 1/2 x 1/2 chance of winning it. As for Arya, he has 1/2 chance of sealing the game by winning the 4th game to make it 3-1, or by winning the decisive 5th game if Bran happens to take the 4th. This would give him the likelihood of winning it by 1/2 + 1/4 ( 1/2 chance of losing 4th game and 1/2 chance of winning the 5th or 1/2 x 1/2). Hence, Arya's likehood of winning the entire event is 3/4 and Bran is 1/4. 3/4 of $120 is $90. Hope am not confusing everyone with my explanation.