Reasons For The Four Results
Let "Bankroll" be the amount of money a gambler has at his disposal at any moment, and let N be any positive integer. Suppose that he raises his stake to when he wins, but does not reduce his stake when he loses. This general pattern is not uncommon among real gamblers, and casinos encourage it by "chipping up" winners (giving them higher denomination chips). Under this betting scheme, it will take at most N losing bets in a row to bankrupt him. If his probability of winning each bet is less than 1 (if it is 1, then he is no gambler), he will eventually lose N bets in a row, however big N is. It is not necessary that he follow the precise rule, just that he increase his bet fast enough as he wins. This is true even if the expected value of each bet is positive.
The gambler playing a fair game (with 0.5 probability of winning) will eventually either go broke or double his wealth. These events are equally likely, or the game would not be fair (ignoring the fact that his bankroll might jump over one event or the other, this is a minor complication to the argument). So he has a 0.5 chance of going broke before doubling his money. Once he doubles his money, he again has a 0.5 chance of doubling his money before going broke. Overall, there is a 0.25 chance that he will go broke after doubling his money once, but before doubling it twice. Continuing this way, his chance of going broke is 0.5 + 0.25 + 0.125 + . . . which approaches 1.
Huygens' result is illustrated in the next section.
The eventual fate of a player at a negative expected value game cannot be better than the player at a fair game, so he will go broke as well.
Read more about this topic: Gambler's Ruin
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