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Best of Donald Catlin

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Sic Bo and Beyond

2 September 2001

Sic Bo is an Asian dice game played with three dice. It is very similar to the proposition bets in the game of craps. Three dice are rolled using either a dice cup or some form of automatic roller. Prior to the roll, the players make wagers on the form of the outcome. For example, one can bet on the total of the three dice, a number 4 through 17 (3 and 18 are triple bets and are elsewhere on the betting layout). One can bet on a specific number 1 through 6 and the payoff is identical to Chuck-a-Luck.

There are a few other bets and most of them are like the proposition bets in Craps, that is to say lousy. So why would I spend a column discussing a game like this? Three reasons. First of all, there is one bet on the layout that is not bad for a proposition bet. Second, there is a variation of Sic Bo that is played in some Canadian casinos which uses a wheel rather than dice and changes the probabilities; I'll discuss this interesting twist later in the article. Finally, and the main reason, is that we can apply our dice calculation techniques from last month's article to the Sic Bo games.

Let me remind you about the techniques in last month's article. Suppose that we want to know the frequency that a total of 10 occurs on the three dice. We first determine what patterns of three numbers, 1 through 6, from highest to lowest, will add up to 10. Here they are in descending order from left to right:

631
622
541
532
442
433

Next we note that there are actually 6 ways that 631 could occur (631, 316, 163, 613, 136, 361) or any other triple for that matter and 3 ways that a double such as 622 could occur (622, 262, 226). Since we have three singles and three doubles, we have 18 + 9 or 27 ways to roll a total of 10. A triple cannot occur here, but if the total were 9 we could have 333 and there is only one way that this can occur. For a more detailed discussion, see last month's article on this site.

Very well, with this simple calculation technique we can easily derive the following table.

Total Frequency
3 or 18 1
4 or 17 3
5 or 16 6
6 or 15 10
7 or 14 15
8 or 13 21
9 or 12 25
10 or 11 27
Total x 2 108 x 2 = 216
Table 1
Dice Total Frequencies for Three Dice

Notice that the total number of different outcomes is 216 which is an independent check of our calculations. Using this table we could determine the house edges for the various wagers on the dice total. For example, a wager on 10 pays 6 to 1. Since there are 27 winners and 189 losers, the house edge is (189 - 27 x 6)/216 or 12.5%. Not very good, is it?

So what about this bet that I said wasn't so bad? It is the bet on a high or low total and pays even money; low totals are 3 through 10 and high totals are 11 through 18. Wait a minute, that sounds like an even money proposition doesn't it? Well yes, were it not for the fact that triples are losers so totals of 3 and 18 are ruled out as are 222, 333, 444, and 555. So, a bet on the High Total, for example, has not 108 winners but only 105, whereas all 108 low totals are losers plus three high triples; altogether 111 losers. In other words, the house edge calculation for a High Total wager looks like this:

Outcome Frequency Payoff Product
High Total 105 +1 +105
Loser 111 -1 -111
Totals - 216 -- -6
Table 2
House Edge Calculation for Sic Bo High Bet

The house edge is just the return to the house per unit wager expressed as a percentage and is

House Edge for High = 6/216 = 2.78% (1)

By symmetry, the house edge for the Low Total bar Triples is exactly this same number.

Now what about this Sic Bo wheel? The Sic Bo wheel is a wheel similar to a Roulette wheel. It has 36 equally spaced compartments and the numbers 1 through 6, represented as spots on a die, are distributed uniformly around the wheel, each number occurring six times. The wheel is spun and three balls are thrown into the wheel and come to rest in three of the 36 cells; no cell can accommodate more than one ball. The outcome of a spin is, as in the dice Sic Bo game, three numbers 1 through 6. At first glance, one might think that the probabilities are the same as for three dice but, as I will show you, this is definitely not the case.

To begin with, since there are 36 cells that the three balls can occupy, the number of outcomes here is the number of ways of choosing a set of three objects from a set of 36 objects. This number, as we have discussed many times before is the binomial coefficient C(36,3) given by

C(36,3) = 36! / 3!(36-3)! = 36 x 35 x 34/6 = 7140 (2)

The set of outcomes is much, much larger than the 216 we had above. Let's see if we can figure out the frequency of sums and calculate the house edge for the high and low total bets.

I will use the total of 9 here since it has all three types of dice patterns, that is, singles, doubles and triples. We can begin just as we did for the dice calculations. Here are the dice combinations in descending order from left to right that add up to 9:

621
531
522
441
432
333

All we have to do now is figure how often a particular triple, double, or three singles occur on the Sic Bo wheel.

Let's begin with 333. There are six of these 3 spots on the wheel and we want the number of ways of landing all three balls in three of them. This number is C(6,3) which is 20. For the combination 441 we figure the number of ways of choosing two 4s from six and one 1 from 6. This number is C(6,2) x C(6,1) and is 90. Finally a combination such as 432 is just C(6,1)3and is 216. We have three triples at 216 each, two doubles at 90 each, and one triple at 20; altogether 648 + 180 + 20 or 848. Similar calculations lead to the following table:

Total Frequency
3 or 18 20
4 or 17 90
5 or 16 180
6 or 15 326
7 or 14 486
8 or 13 702
9 or 12 848
10 or 11 918
Total x 2 3570 x 2 = 7140
Table 3
Sum Frequencies For the Sic Bo Wheel

Here a wager on the 10 at 6 to 1 has a house edge of 10%, a bit better than in the dice game.

For the High wager we have the following. There are, for example, 3570 high totals but 60 of them are triples and these bets lose. Thus there are 3510 winners and 3630 losers. Here is the house edge table for the High Total bet.

Outcome Frequency Payoff Product
High Total 3510 +1 +3510
Loser 3630 -1 -3630
Totals - 7140 -- -120
Table 4
House Edge Calculation for Sic Bo Wheel High Bet

The house edge here is just

House Edge for High Wager = 120/7140 = 1.68% (3)

Once again, by symmetry, the house edge for the Low wager is the same as that for the High wager. Note that both wagers are considerably better on this wheel game than they were in the dice game. In general, the wagers on the wheel game are better than the dice game. For example, the single number wager (the one that pays like Chuck-a-Luck) has a house edge of 7.87% in the dice game and 6.86% on the wheel.

Now here is something to ponder. If the only calculation I was interested in making here was that for the High and Low totals bar triples, I could have calculated the house edge in both Sic Bo versions by simply knowing the number of outcomes in each game and the total number of triples in each game. A glance at Tables 2 and 4 will show you why. If you have questions, I can be reached via email by clicking on the word Technigame in the 'About the Author' box at the end of this article.

Next month I am going to present an interesting calculation that reveals a powerful fact about playing the comp game. I think you won't want to miss this one. See you then.

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