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Best of Donald Catlin

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Calculating the House Edge in Let It Ride

3 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.

3 Card Hand Number of Expected Return Product
Royal Double Inside 24 1172 28.128
Royal Single Inside 12 1454 17,448
Royal Outside 4 1736 6,944
Playable SF 0 high 20 252 5,040
SF 1 high 4 480 1,920
SF 2 high 4 708 2,832
SF Single Inside 1 high 12 198 2,376
SF Single Inside 2 high 12 426 5,112
SF Double Inside 2 high 24 144 3,456
High Pair (T - A) 1440 1689 2,432,160
3 of a Kind 52 6360 330,720
Subtotal - 1608 ---- 2,836,136
Folds 20,492 0 0
Total 22,100 ---- 2,836,136
Table 1
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.

4-Card Hand Number of Expected Return Product
Royal Inside 16 1067 17,072
Royal Outside 4 1277 5,108
SF Inside 0 high 48 243 11,664
SF Inside 1 high 32 249 7,968
SF Inside 2 high 16 255 4,080
SF Inside 3 high 16 261 4,176
SF Outside 0 high 20 453 9,060
SF Outside 1 high 4 459 1,836
SF Outside 2 high 4 465 1,860
SF Outside 3 high 4 471 1,884
Flush 0 high 212 33 6,996
Flush 1 high 1084 39 42,276
Flush 2 high 1100 45 49,500
Flush 3 high 300 51 15,300
Totals - 2860 ---- 178,780
Table 2
Expected Return on 4-Card Flush Hands

Here are the non-Flush hands that are relevant to optimum strategy:

4 Card Hand Number of Expected Return Product
Straight Outside 0 high 1,260 0 0
Straight Outside 1 high 252 6 1,512
Straight Outside 2 high 252 12 3,024
Straight Outside 3 high 252 18 4,536
Straight Outside 4 high 252 25 6,048
Straight Inside 4 high 1,008 0 0
High Pair (T - A) 31,680 58 1,837,440
Two Pair 2,808 132 370,656
3 of a Kind 2,496 215 536,640
4 of a Kind 13 2400 31,200
Subtotal - 40,273 ---- 2,791,056
Flush Totals 2,860 ---- 178,780
Folds 227,592 0 0
Grand Total - 270,725 ---- 2,969,836
Table 3
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.

5 Card Hand Number of Payoff Table Product
Royal 4 1000 4,000
Straight Flush 36 200 7,200
4 of a Kind 624 50 31,200
Full House 3,744 11 41,184
Flush 5,108 8 40,864
Straight 10,200 5 51,000
3 of a Kind 54,912 3 164,,736
Two Pair 123,552 2 247,104
High Pair 422,400 1 422,400
Low Pair 675,840 -1 -675,840
High Card 1,302,540 -1 -1,302,540
Totals - 2,598,960 ---- -968,692
Table 4
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.

Hand Number Played Return
1 1,891,008 2,836,136
2 4,140,768 5,939,672
$ 25,989,600 -9,686,920
Totals - 32,021,376 -911,112
Table 5
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.

Donald Catlin

Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers
Donald Catlin
Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers