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The Mathematics of Sic Bo

1. The Game Of Sic Bo. Sic Bo, also known as Tai Sai or Big Small, is an ancient Chinese game of chance played with three dice. It is presently one of the popular casino games in Asia and it has been gradually gaining popularity among westerners too. It is easy to play and it offers many types of bet with payout as high as 195 to 1.
2. Sic Bo is a very popular dice game in the United States. It is played at many of the land-based casinos in Sin City. However online casinos mean that you never have to travel to one of those casinos to enjoy Sic Bo. All best-rated online gambling sites offer players the chance to play the Chinese game online.

See full list on gambledex.com. Sic Bo (pronounced See Bo) literally means 'dice pair', but the game is actually played with three dice. It is also known as Tai Sai (Lucky dice) and Dai Siu (Big Small). A number of English games have been developed on the same lines as Sic Bo, namely Grand Hazard and Chuck-a-Luck.

By

Michael Shackleford

January 21, 2005

Sic Bo, meaning 'dice pair' is an ancient Chinese gambling game. Today it is one of the lesser known casino games and is often confined to designated rooms for Asian games. The game uses three dice and a table with a variety of betting options on the roll of those dice. The odds and table layout may also vary from place to place. If you must play Sic Bo I would suggest sticking to only the 'low' and 'high' bets.

Following is a list of the bets available. The payoffs listed are for Atlantic City and the Mirage in Las Vegas. Other casino’s odds will vary.

• Small: Wins on total of 4-10, except for a three of a kind. Pays 1 to 1.
• Big: Wins on total of 11-17, except for a three of a kind. Pays 1 to 1.
• 4: Wins on total of 4. Pays 60 to 1.
• 5: Wins on total of 5. Pays 30 to 1.
• 6: Wins on total of 6. Pays 17 to 1.
• 7: Wins on total of 7. Pays 12 to 1.
• 8: Wins on total of 8. Pays 8 to 1.
• 9: Wins on total of 9. Pays 6 to 1.
• 10: Wins on total of 10. Pays 6 to 1.
• 11: Wins on total of 11. Pays 6 to 1.
• 12: Wins on total of 12. Pays 6 to 1.
• 13: Wins on total of 13. Pays 8 to 1.
• 14: Wins on total of 14. Pays 12 to 1.
• 15: Wins on total of 15. Pays 17 to 1.
• 16: Wins on total of 16. Pays 30 to 1.
• 17: Wins on total of 17. Pays 60 to 1.

• Two of a kind: Player may bet on any of the 15 possible two dice combinations (for example a 1 and 2). Bet wins if both numbers appear. Probability of winning is 13.89%. Pays 5 to 1.
• Double: Player may bet on any specific number (for example a 1). Player wins if at least 2 of the 3 dice land on that number. Probability of winning is 7.41%. Pays 10 to 1.
• Triple: Player may bet on any specific number (for example a 1). Player wins if all 3 dice land on that number. Probability of winning is 0.46%. Pays 180 to 1.
• Any Triple: Wins on any three of a kind. Pays 30 to 1.
• Individual Number: Player may bet on any specific number from 1 to 6. If chosen number appears 1 time bet pays 1 to 1, if it appears 2 times bet pays 2 to 1, and if it appears 3 times it pays 3 to 1.

The critical step in calculating the odds in Sic Bo is to find the probability of any given total in the throw of three dice. Following is a formula for s spots over n dice, taken from The Theory of Gambling and Statistical Logic by Richard A. Epstein, formula 5-14.

For example, let's look at the number of ways to get 11 spots over 3 dice.

int[(s-n)/6] = int[(11-3)/6] = int[1.33] = 1

The total would be 6^-3 * [-1⁰*combin(3,0)*combin(11-6*0-1,3-1) + -1¹*combin(3,1)*combin(11-6*1-1,3-1) ] =
1/218 * [1*1*combin(10,2) + -1*3*combin(4,2)] =
1/218 * [1*1*45 + -1*3*6] =
1/218 * [45-18] = 27/216 = 12.50%

Alternatively, if you can program a computer that would probably be the fastest way to get the results.

Here is a simple function in C++.

Following is the output of the function.

If you don’t know how to program you’re going to have to do this the hard way. What I recommend is list every combination of 3 dice. To avoid the list being 6³=216 items long do not repeat the same combinations in different orders. In the interests of not listing the same number twice always order each combination from lowest to highest, not forgetting combinations with a pair or three of a kind.

So we start with 1,1,1.

Next would be 1,1,2.

Then 1,1,3; 1,1,4; 1,1,5; and 1,1,6.

Obviously you can’t roll a 7 with one dice so next we increment the second die.

1,2,?

The third die must be greater or equal to the second die so the next combination in full would be 1,2,2.

Next comes 1,2,3; 1,2,4; 1,2,5; and 1,2,6.

Then comes 1,3,3.

I hope you see the pattern. The whole list would look like the following.

Next we have to determine the number of permutations of each combination. A combination is a set without regard to order and a permutation is a set with regard to order.

With a three of a kind there is only one way permutation. For example if the three dice are 1,1,1 there is only one way to roll that a 1 each time.

If the combination is 1,1,2 there are three ways to roll that: 1,1,2; 1,2,1; and 2,1,1.

If all three dice are 1,2,3 there are six possible permutations: 1,2,3; 1,3,2; 2,1,3; 2,3,1; 3,1,2; 3,2,1

The general formula is that if you have a total of d dice and the totals of each number are x₁, x₂, x₃…x_n then the number of permutations are d!/(x₁!*x₂!*x₃*…*x_n). So the number of ways to get a three of a kind would be 3!/3! = 6/6 = 1. The number of ways to get a pair would be 3!/(2!*1!) = 6/(2*1) = 3. The number of ways to get three different numbers would be 3!/(1!*1!*1!) = 6/(1*1*1) = 6.

Next we go through the tedious process of adding the number of permutations for each total. For example a total of 6 has the following combinations with the corresponding number of permutations.

The final table will look like this, not unlike the result of the computer function earlier.

So now lets add a column to our list for the number of combinations of each set. Let’s also add a total for the three dice.

Now we can divide each total number permutations by the total number 3-dice permutations (216) to get the probability of each total.

Finally, we are ready to evaluate the expected value of each bet. The expected value is the ratio of the amount the player can expect to win to the amount he bets on any given bet. So a fair bet would an expected value of zero. A positive expected value would mean the player has the advantage. A negative expected value would mean the dealer has the advantage.

Let’s start with the 4 bet. This wins with a total of 4 and pays 60 to 1. For those who don’t know, '60 to 1' means if the player wins he wins 60 times his bet and KEEPS his original wager. Had the odds paid '60 for 1' the player would NOT keep his original bet. Most table games pay on a 'to 1' basis.

The probability of a total of 4 is 3/216 = 0.013889. Thus the probability of losing is 1-(3/216) = 1-0.013889 = 0.986111.

The expected value of any bet with only two possibilities, winning or losing, is:

(Probability of winning)*(Amount of win) + (Probability of losing)*(Amount of loss).

For the 4 bet the expected value is

0.013889 * 60 - 0.986111*-1 = -0.15278.

So, this tells us that for every dollar the player bets on a total of 4 he can expect to lose 15.278 cents on average. Or, the house edge is 15.278%.

The next table shows the expected value and how it was calculated for all bets of a total of 4 to 17.

Two of a kind

There are combin(6,2)=6!/(4!*2!)=15 ways to choose two numbers out of six. Each of these combinations is listed on the table and the player bet on as many as he wishes. If both numbers appear on the roll of the three dice then the player wins and is paid 15 to 1.

Let’s assume the player picks a 1 and 2 as his two numbers. What is the probability that both a 1 and 2 occur in the roll of 3 dice? One way to do this would be to note all the possible winning permutations:

Thus there are a total of 30 winning permutations.

There are 6³=216 total permutations, so the probability of winning is 30/216 = 1/36 = 0.1388889

The two of a kind bet pays 5 to 1. So the expected value is 0.1388889*5 + (1-0.1388889)*-1 = -0.16667. In other words the house edge is 16.67%.

Double

There are six double bets available, one for each number from 1 to 6. The player may be on any one or combination of bets. Any given bet wins if at least two of the three dice land on that number.

Let’s assume the player bets on the 1.

One way to solve it would be to note all the winning permutations:

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Thus there are a total of 16 winning permutations.

There are 6³=216 total permutations, so the probability of winning is 16/216 = 0.0740741.

The double bet pays 10 to 1. So the expected value is 0.0740741*10 + (1-0.0740741)*-1 = -0.18518. In other words the house edge is 18.52% (ouch!).

Triple

Player may bet on any specific number (for example a 1). Player wins if all 3 dice land on that number.

There is obviously only one way to win this bet, so the probability of winning is 1/216 = 0.0046296. The bet pays 180 to 1 so the expected value is 0.0046296*180 + (1-0.0046296)*-1 = -0.16204. So the house edge is 16.204%.

Any Triple

The Any Triple bet pays if any three of a kind is thrown. There are obviously six winning combinations (1,1,1; 2,2,2; 3,3,3; etc.). So the probability of winning is 6/216 = 0.027778. The bet pays 30 to 1 so the expected value is 0.027778*30 + (1-0.027778)*-1 = -0.13889. So the house edge is 13.89%.

Low

The low bet wins if the total of the three dice is 3 to 10, without being a three of a kind. The probability of any total 10 or less is exactly 50%. The average number on any one die is (1+2+3+4+5+6)/6 = 21/6 = 3.5. So the average of three dice is 3*3.5 = 10.5. It stands to reason that the probability of getting under or over 10.5 is 50%.

However the bet loses on a three of a kind. There are 3 three of a kinds that would turn a winner into a loser: 1,1,1; 2,2,2; and 3,3,3. So the probability of having a total of 10 or less as a three of a kind is 3/216 = 0.0188889. So the overall probability of winning is 0.5 – 0.188889 = 0.4861111. The bet pays 1 to 1 so the expected value is 0.4861111*1 + (1-0.4861111)*-1 = -0.02778. Thus the house edge is 2.78%.

High

The high is just the opposite of the low bet, so it stands to reason the house edge would also be 2.78%.

Individual Number

Player may bet on any specific number from 1 to 6. If chosen number appears 1 time bet pays 1 to 1, if it appears 2 times bet pays 2 to 1, and if it appears 3 times it pays 3 to 1. Probability of 1 match is 34.72%, 2 matches is 6.94%, 3 matches is 0.46%.

Let’s assume the player picks the number one.

There is only one way to get three ones: 1,1,1. So the probability of three ones is 1/6³ = 1/216.

Following are the ways to get two 1’s and the number of permutations of each.

So the probability of two ones is 15/6³ = 15/216.

Following are the ways to get one 1 and the number of permutations of each.

So the probability of two ones is 75/6³ = 75/216.

Another way to arrive at the probability of one 1 would be find the probability that the first die is a one and the second and third are not:

Pr(one)*Pr(not one)*Pr(not one) = (1/6)*(5/6)*(5/6) = 25/216.

However the one could appear in the first, second, or third position, so multiply by 3: 3*(25/216) = 75/216.

The probability of rolling zero ones is Pr(not one)*Pr(not one)*Pr(not one) = (5/6)*(5/6)*(5/6) = (5/6)³ = 125/216.

The following return table shows the possible outcomes, and the number of combinations, probability, and return of each. The return is the product of the probability and the win or loss to the player.

So the total expected return is -0.0787, or the house edge is 7.87%.

Sic Bo section at the Wizard of Odds.
How to calcualte the 3-dice permutation in Visual Basic.

Table Of Contents

• Sic Bo is an ancient Chinese gambling game
• It is played with three dice
• The Sic Bo table is very similar to the one of Roulette

With this article you are going to learn how to play and get the best odds to win at Sic Bo - a Casino game most people know nothing about.

If you never heard of Sic Bo, I’m not surprised.

Slots, Blackjack, Roulette, Video Poker, and Baccarat are awesome Casino games. No argument there.

And they are everywhere. From Casinos, obviously, to references in pop culture. People know about these games even if they have never played any of them.

But that’s not all Casinos have to offer. If you checked every single game even at the smallest online Casino, you’d find something you never heard about.

Like the game of Sic Bo.

Sic Bo in a Nutshell

What is Sic Bo? It’s a game of chance played with three dice. It’s originally from China and sometimes gets referred to as Dai Siu, Tai Sai, Hi-Lo, or the Big and Small game.

Did you know there are only two Casino games including dice? And Sic Bo is one of them.

The game has a couple of variants, including an English version called Grand Hazard and the US version Chuch-a-luck.

Sic-bo Dice Casino

Did you know that you can play Sic Bo online? Start playing now - both for free and for real money!

_{18+. Gamble responsibly.}

The original Sic Bo game is most popular in Asia and Macau Casinos. But with the power of the internet, you can play it online no matter where you are.

And this Sic Bo guide will help you to start playing the game. And to avoid common mistakes that beginners make.

How to Play Sic Bo Online

The games of chance are easy to learn. Usually. But if you take a look at Sic Bo table for the first time in your life, you will feel very lost.

If you thought Craps and Roulette table layout was complicated, you should look at this beauty.

It may be frustrating at first but it’s not that difficult to learn the Sic Bo rules. And it’s even easier when you play online. You’ll see in this guide.

The main goal of this game is not that different from other Casino games. It involves betting on specific dice outcomes.

The gameplay goes like this:

• you place your bets by putting chips on areas depicting outcomes you want to bet on
• the dealer shakes three dice closed in a tiny chest (or the game software “rolls” the dice and shows their random outcomes)
• you win for every number or sum you have predicted and lose if your predictions were wrong

Sic Bo Betting Options

As you can guess, the betting options are the important (and fun) part of the game. And you can see from the Sic Bo table, there are plenty of those.

Small and Big Bets

The most basic ones you need to know are the Small and the Big bets.

• The Small bet is when you bet on any dice sum between (and including) 4 and 10.
• The Big bet is when the sum is between (and including) 11 and 17.

They usually pay 1 to 1 and have the smallest house edge. Thus, these are the best odds you can have in a Sic Bo game.

But you may notice the sums of 3 and 18 are missing. That’s because when you bet Small or Big, any triple sum (three dice with same numbers, e.g. 1-1-1, 6-6-6) makes you lose.

Thus, these bets are not exactly 50/50 but have a probability of 48.61 percent.

Other Sums Bets

Don’t feel like betting on specific dice numbers and their combinations? You can always choose one specific sum and bet on it.

It can be anything between (and including) 4 and 17.

But different sums have a different amount of winning possibilities.

For example, you can get the sum of 4 only three ways: 1-1-2, 1-2-1, and 2-1-1. Same goes for 17. Thus, these sums have the biggest payouts and the biggest house edge.

You can see the payouts of sum bets in the table below:

Single Dice Bet

If you make Single Dice Bet, you predict that a specific number will appear on one, two, or all three dice.

You can choose any single number - from 1 to 6. If it appears on one dice, the payout is 1 to 1. If it appears on two dice - 2 to 1, on three - 3 to 1.

Double and Triple Bets

You will want to make a Double Bet when you have a strong feeling that two of the dice will come up having the same number.

This bet pays 10 to 1 but you have only 7.4 percent chance to predict it right.

Like the double, there are triple bets, too. You can bet that all three dice will have the same numbers. And if you are right, it pays 30 to 1.

You can also bet on a specific set of triples — but your odds are worse here.

Two Dice Combination Bets

If you look further Sic Bo table layout, you’ll see 15 domino-like areas for different two dice combinations, e.g. 1-2, 2-3, 3-5, etc.

That’s another thing you can bet on. Sometimes it’s called Domino bet, or simple Two Dice Combination bet.

It predicts the outcomes of two dice at once rather than all three or their sum. This bet pays 5 to 1.

Betting Options Recap

To make it even less confusing, here’s a table with all the bets, payouts, possible winning combinations, and house edge.

Make sure to use if the first times you play Sic Bo. Later on, you’ll get a hold of your odds for different bets and have a better understanding of which bets are better to make.

How to Get the Best Chances to Win at Sic Bo Casino Game

Finding ways how to play optimal Sic Bo may be what led you to this article in the first place.

Then I have good news and bad news for you.

Bad news first: there is not a single solid Sic Bo betting strategy that would help you to win every single time.

I know, I know, that’s unfortunate.

But it’s a game of chance, after all. And there is nothing you can do to predict the exact outcomes of the dice.

Yet (the good news) if you want to play better get the best chances to win that the game allows, you CAN count on some Sic Bo strategy tips.

1. Don’t Get Tempted to Bet on Too Many Areas

When playing Sic Bo online, you can place multiple bets at once. Online Casinos usually allow betting on up to 16 areas per one roll. But I wouldn’t recommend betting on that many.

Sure, you’ll win multiple bets. But how many will you lose in the process?

Instead, analyze all the betting options.

• What are the chances of winning for each of them?
• Are the payouts even worth it?

Then you’ll have your own Sic Bo strategy. And you'll manage your bankroll better than when playing the game randomly.

2. Place Most Bets on Big or Small Area

This may be an obvious choice by now. The Big and Small bets have the best odds and the smallest house edge.

If you play at land-based Casino, you can find some players betting only on Big and Small bets.

But betting only on those would be pretty boring, don’t you think?

And the fun element in Casino games is more important than the prizes you win. Thus, don’t forget to explore other betting options while still keeping most (and highest) bets on the Big or Small areas.

3. Don’t Fall for the Gambler’s Fallacy

Gambler’s fallacy is a common myth in gambling. It states that if something happens more frequently at some point, it will occur less frequently later on.

As in if the dice have been rolling Big sums for a long time, it will definitely roll a lot of Small ones now.

False.

The outcomes of the dice are not dependent on what happened before. So, don’t follow this blindly. Not in Sic Bo, not in Craps, and not in any other game of chance.

Where to Play Sic Bo Online

This guide was explicit enough, so don’t panic - I won’t bore you with hundreds of places where to play Sic Bo.

I doubt there are that many places online anyway. And even if there are, only a few would be any good for playing Sic Bo.

My recommendation?

No matter whether you’re a beginner or already had experience playing Sic Bo, you should try the games at this online Casino.

It features two Sic Bo platforms:

BGAMING’s platform may be a bit simpler and clearer, so you won’t get confused even if it’s your first time playing the game.

All the payouts are written on the layout already. And you can use the same platform for both free and real money gambling.

Open a FREE account and play Sic Bo for free.For as long as you like.

Bonus, you can change the color of the Sic Bo table. Maybe you have a lucky color you’d prefer to use?

If you want to feel the Chinese roots while playing the game, go for a Habanero platform.

It’s more traditional and has that Oriental feel to it.

But when it comes to the payouts, house edge, or any other important numbers, you’ll have the same experience no matter which Sic Bo platform you choose for playing.

祝好运 / Good luck!