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.
Player $A$ wins in the following situations:
- In the next two flips, both are heads
- In the next three flips, you get a head and a tail in some order, followed by heads
- 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.