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A coin is tossed until it gives either $10$ heads or $10$ tails. Player $A$ bets on $10$ heads and player $B$ bets on $10$ tails. The game is unexpectedly interrupted after $15$ tossings with $8$ heads and $7$ tails observed. What would be the fair ratio to split the prize pool between player $A$ and $B$? Consider it be the ratio of winning probabilities of the players.

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    $\begingroup$ Are you aware that this was historically an important problem in the development of probability theory? :-) en.wikipedia.org/wiki/Problem_of_points $\endgroup$ – joriki Feb 14 at 19:54
  • $\begingroup$ Welcome to MSE. Please include your own thoughts and the effort made thus far, so that people can work with you accordingly. (Please add those in the body of the question instead of commenting.) $\endgroup$ – Lee David Chung Lin Feb 14 at 21:32
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Hint:

Player $A$ wins in the following situations:

  1. In the next two flips, both are heads
  2. In the next three flips, you get a head and a tail in some order, followed by heads
  3. In the next four flips, you get a head and two tails in some order, followed by heads

Those are the only ways that $A$ will win. Figure out the probability of those three cases, and player $B$ has the complementary probability.

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