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Calculating the House Edge in Let It Ride3 June 2001
In last month's article I showed you how to derive the optimum strategy for playing the casino game Let it Ride. We did this with surprisingly few calculations. As I promised in that article, this month I am going to show you how to compute the house edge given that the player uses the strategy we developed. This, unfortunately, takes quite a few calculations. All of these calculations are the same or similar to the calculations we did last month, so I am not going to drag you through that again. Rather, I'll simply report the results of the individual calculations, and if you have questions you can contact me via email by clicking on the word 'Technigame' in the About the Author box at the end of the article.
Recall that there are three betting spots in Let it Ride and the first one involves making a play/fold decision upon seeing three cards. This means that there are C(52,3) or 22,100 different hands to play. Each of these can be completed in C(49,2) or 1,176 different ways. Altogether then, there are 1,176 x 22,100 or 25,989,600 different scenarios. We'll come back to this number later. Using the three-card strategy we developed last month, we can construct the following table. A few of these numbers you may recognize from last month.
Expected Return on 22,100 Three-Card Hands
Notice that most of the time the three-card hand will be played because of a high pair. The figure 2,836,136 represents the amount of money one would expect if each of the 22,100 starting hands were completed in each of the 1,176 ways (including folds).
The four-card hands have a similar table, except that it is considerably larger. For this reason I am going to break it into two tables, one for Flush hands and one for non-Flush hands. Note that there are C(52,4) or 270,725 different four-card hands.
Expected Return on 4-Card Flush Hands
Here are the non-Flush hands that are relevant to optimum strategy:
Expected Returns for Four-Card Hands using Optimum Strategy
The number 2,969,836 is the number of units returned to the player if each of the 270,725 four-card hands is played using optimum strategy, assuming each of the 48 ways to fill the fifth card occurs (including folded hands).
Finally we have the five-card $ hand. This hand cannot be folded, so there are no decisions to make. We simply compute how many times each type of hand occurs and calculate the expected return using the payoff table. There are C(52,5) or 2,598,960 5 card hands. Here goes.
Expected Returns for Five-Card $ Hand
Next note that there are 22,100 x 1,176 or 25,989,600 hands that we must cycle through to get the expected return for Hand 1, 270,725 x 48 or 12,994,800 hands that we must cycle through to get the expected return for Hand 2, and only 2,598,960 hands to get the expected return for the $ Hand. The largest of these is the number for Hand 1. To get proper comparisons we will use this number of hands for Hand 2 and the $ Hand. This means that the numbers in Table 3 have to be multiplied by 2 and those in Table 4 have to be multiplied by 10. Doing so gives us the following summary table.
Summary of Results
The number played column is the actual number of the 25,989,600 hands that are played for each type of hand, that is, these are non-folded hands. If one unit is wagered on each hand, then the number 32,021,376 is the total amount actually wagered. This means that the average bet using optimum strategy is approximately 1.2321 units. The figure -911,112 is the return to the player and the negative sign indicates that it is a loss. The return per game is -911,112 divided by 25,989,376 and is approximately -0.035057. The house edge is 911,112 divided by 32,021,376 and is approximately 2.8453%. Had we folded the outside Straights with 0 high cards and the inside Straights with 4 high cards, the edge would be just slightly higher at 2.865% since the average bet would be lower. Usually, Let it Ride is reported as having a house edge of 2.85%, which seems like a reasonable figure since some people will play these hands and others won't.
So much for Let it Ride; I think we've beaten this one to death. 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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