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Puzzle – Gambler’s Ruin

Which is better – wealth or ability?  Fred Mosteller posed this question in his classic 1965 small compendium Fifty Challenging Problems in Probability, in the context of the Gambler’s Ruin puzzle. 

Mosteller book

Two players, M and N, engage in a game in which $1 is transferred from one player to the other at each play. Player M has a ⅔ probability of winning each play (more “ability”), but starts out with only $1 (less “wealth”).  Player N has a ⅓ probability of winning and starts with  $2.  Obviously, player M has a ⅓ chance  of going bankrupt immediately, but in the ⅔ of cases where they win the first play, does M survive long enough for their superior ability to give them the better chance of escaping long-term bankruptcy?  

The answer is yes – ultimately M has a 4/7 chance winning the game (put another way, N has a 4/7 chance of going bankrupt).  Mosteller derived this probability from a more general solution for player M’s probability P of winning the game:

P =  [1 – (q/p)m] / [1 – (q/p)(m+n)]


m is M’s initial stockpile
n is N’s initial stockpile
p is M’s probability of winning a single play of the game
q is N’s probability of winning a single play of the game

Conclusion (in Mosteller’s words):  “It’s better to be twice as good a player than twice as wealthy.”