On games 1 and 2, bet $31.25. On game 3, bet $25.00 if your team is 2-0 or 0-2, and $37.50 if the record is 1-1. On game 4, bet $12.50 if the record is 3-0 or 0-3, and bet $37.50 if the record is 2-1 or 1-2. For game 5 (if you get to it), bet $25 if the record is 3-1 or 1-3, or $50 if the record is 2-2. If there is a game 6, bet $50 on it. If you get to a game 7 (so the record is 3-3), bet $100.
Here is why this works. At game 7, you need to have had $0 profit or loss from the previous games, so you want to bet $100 to replicate your winner-take all bet. Taking a step back, this means that if your team was 3-2, you would have had to lose enough money to get to $0 profit, and win $100 if they won, so your bet would have had to be $50. (Same on the losing side, so you would lose $100 if your team lost and they were 2-3 coming in to the game, or gotten back to $0 if they won to get to 3-3.) The reverse induction works all the way back to the initial bet of $31.25.
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Today's brain teaser courtesy of Braingle.com.