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Some Gambler's Ruin Questions1 May 2005
One of my readers, Ralph Scalitzia, sent me the following email after an earlier response to one of his questions:
Wow, Ralph, you have lots of questions here. Let me start with the last one first. The short answer is no! Let me explain. The 1.35% house edge figure you state is for European Roulette (single zero) with the added feature of the en prison rule. So I'll first explain the en prison rule and then we'll determine the correct probabilities.
If one bets on (say) Red and the ball lands on a red number, the player wins even money. If the ball lands on a black number, the player loses his wager. On the other hand, if the ball lands on the zero then the player neither wins nor loses and the player's bet is carried over to the next spin - the bet is said to be "in prison." On the next spin if the ball lands on a red number then the player's wager is returned. If the ball lands on a black number, the player's wager is collected. If the zero occurs again the bet remains en prison and the wager is carried over to the next spin. This procedure continues until the bet is settled.
First of all, note that there are three outcomes here: a win, a loss or a tie. The player can only win on the first spin and that probability is 18/37 or approximately 0.4864865 (hence my no response to your question). Likewise there is an 18/37 chance of losing on the first spin. The probability of the zero occurring is 1/37. Of course, it could occur again, and again, ad infinitum. So as you might expect there is an infinite geometric series lurking in the background here. Although one can analyze the problem using the series, there is a simple observation one can make that avoids that analysis. Simply put, once the first zero occurs, there is an even chance of subsequently either losing or obtaining a tie. Thus the probability of tying is one half of 1/37 or 1/74. Likewise, the probability of losing after the initial spin is also 1/74. Thus the probability of losing is 36/74 + 1/74 or 37/74 (exactly ½). In summary, we have the following table
1/74 expressed as a percentage is approximately 1.35135%.
Now I'll address your Case (1) question. I'll assume that you are playing a single zero Roulette game with no en prison feature. Although some Atlantic City games have a Surrender feature (different from en prison), I know of no en prison games in the U.S. Thus the correct win probability is not 0.493 but rather 18/37 or approximately 0.4865 (the same as it is for the en prison game by the way); losing probability is 19/37 or approximately 0.5135 (in the en prison game it is 1/2).
The standard formula for Gambler's Ruin in such a game is determined as follows. If p is the probability of winning one unit and q is the probability of losing one unit we define
f(x) = (q/p)x (1)
Then the ruin probability is given by the formula
r(z) = [f(a) - f(z)]/[ f(a) - 1] (2)
where z represents the player's current stake and a represents the player's goal. Note that r(a) = 0 (no chance of ruin if the goal is reached) and r(0) = 1 (ruin is certain). For the problem at hand (q/p)50 =19/18. Hence (q/p)50 = 14.92983 and (q/p)51 = 15.75926. The difference here is 0.82943 and the ruin probability according to the formula in (2) is 0.82943/14.75926 or 0.056197. By the way, the formulas in (1) and (2) apply to the en prison game as well, but the probabilities used must be those in Figure 1, that is q/p = 37/36.
You may wonder how one arrives at the formulas in (1) and (2). Next month I'll address Ralph's Case (2) question in detail and you'll then see the techniques used in deriving these ruin formulas. Not all of them are tractable but fortunately the question Ralph raised in Case (2) can be answered (though not without some struggle). See you next month.
This article is provided by the Frank Scoblete Network. Melissa A. Kaplan is the network's managing editor. If you would like to use this article on your website, please contact Casino City Press, the exclusive web syndication outlet for the Frank Scoblete Network. To contact Frank, please e-mail him at firstname.lastname@example.org.
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