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12.2: Gambler's Ruin
https://stats.libretexts.org/Bookshelves/Probability_Theory/Introductory_Probability_(Grinstead_and_Snell)/12%3A_Random_Walks/12.02%3A_Gambler's_Ruin
With the new values, the expected fortune of player A (that is, the sum of the nominal values of his counters) after each play equals his fortune before the play (and similarly for player B). (For a s...With the new values, the expected fortune of player A (that is, the sum of the nominal values of his counters) after each play equals his fortune before the play (and similarly for player B). (For a simpler martingale argument, see Exercise [exer 12.2.10].) De Moivre then uses the fact that when the game ends, it is still fair, thus Equation \(\PageIndex{1}\) must be true.

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