Sic Bo mathematics

(Source: http://www.mathproblems.info/gam470/games/sicbo/)

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++.

void SicBo(void)
{
int i,d1,d2,d3,tot,tot_array[19];
for (i=0; i<=18; i++)
tot_array[i]=0;
for (d1=1; d1<=6; d1++)
{
for (d2=1; d2<=6; d2++)
{
for (d3=1; d3<=6; d3++)
{
tot=d1+d2+d3;
tot_array[tot]++;
}
}
}
cout << "Total\tCombinations\tProbability\n";
for (i=3; i<=18; i++)
cout << i << "\t" << tot_array[i] << "\t" << (float)tot_array[i]/216 << "\n";
}

Following is the output of the function.

Total

Permutations

Probability

3

1

0.00463

4

3

0.013889

5

6

0.027778

6

10

0.046296

7

15

0.069444

8

21

0.097222

9

25

0.115741

10

27

0.125

11

27

0.125

12

25

0.115741

13

21

0.097222

14

15

0.069444

15

10

0.046296

16

6

0.027778

17

3

0.013889

18

1

0.00463



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.

Low die

Medium Die

High Die

1

1

1

1

1

2

1

1

3

1

1

4

1

1

5

1

1

6

1

2

2

1

2

3

1

2

4

1

2

5

1

2

6

1

3

3

1

3

4

1

3

5

1

3

6

1

4

4

1

4

5

1

4

6

1

5

5

1

5

6

1

6

6

2

2

2

2

2

3

2

2

4

2

2

5

2

2

6

2

3

3

2

3

4

2

3

5

2

3

6

2

4

4

2

4

5

2

4

6

2

5

5

2

5

6

2

6

6

3

3

3

3

3

4

3

3

5

3

3

6

3

4

4

3

4

5

3

4

6

3

5

5

3

5

6

3

6

6

4

4

4

4

4

5

4

4

6

4

5

5

4

5

6

4

6

6

5

5

5

5

5

6

5

6

6

6

6

6

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…xn then the number of
permutations are d!/(x1!*x2!*x3*…*xn). 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.



Low die

Medium die

High die

Total

Permutations

1

1

1

3

1

1

1

2

4

3

1

1

3

5

3

1

1

4

6

3

1

1

5

7

3

1

1

6

8

3

1

2

2

5

3

1

2

3

6

6

1

2

4

7

6

1

2

5

8

6

1

2

6

9

6

1

3

3

7

3

1

3

4

8

6

1

3

5

9

6

1

3

6

10

6

1

4

4

9

3

1

4

5

10

6

1

4

6

11

6

1

5

5

11

3

1

5

6

12

6

1

6

6

13

3

2

2

2

6

1

2

2

3

7

3

2

2

4

8

3

2

2

5

9

3

2

2

6

10

3

2

3

3

8

3

2

3

4

9

6

2

3

5

10

6

2

3

6

11

6

2

4

4

10

3

2

4

5

11

6

2

4

6

12

6

2

5

5

12

3

2

5

6

13

6

2

6

6

14

3

3

3

3

9

1

3

3

4

10

3

3

3

5

11

3

3

3

6

12

3

3

4

4

11

3

3

4

5

12

6

3

4

6

13

6

3

5

5

13

3

3

5

6

14

6

3

6

6

15

3

4

4

4

12

1

4

4

5

13

3

4

4

6

14

3

4

5

5

14

3

4

5

6

15

6

4

6

6

16

3

5

5

5

15

1

5

5

6

16

3

5

6

6

17

3

6

6

6

18

1

Total

216

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.

Combinations

Number of Permutations

1,1,4

3

1,2,3

6

2,2,2

1

Total

10

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.

Total

Permutations

3

1

4

3

5

6

6

10

7

15

8

21

9

25

10

27

11

27

12

25

13

21

14

15

15

10

16

6

17

3

18

1

Total

216

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

Total

Permutations

Probability

3

1

0.00463

4

3

0.013889

5

6

0.027778

6

10

0.046296

7

15

0.069444

8

21

0.097222

9

25

0.115741

10

27

0.125

11

27

0.125

12

25

0.115741

13

21

0.097222

14

15

0.069444

15

10

0.046296

16

6

0.027778

17

3

0.013889

18

1

0.00463

Total

216

1

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.

Total

Pays

Probability of Win

Probability of Losing

Formula of expected value

Expected Value

4

60

0.013889

0.986111

0.0138889*60-0.986111*-1

-0.15278

5

30

0.027778

0.972222

0.0277778*30-0.972222*-1

-0.13889

6

17

0.046296

0.953704

0.0462963*17-0.953704*-1

-0.16667

7

12

0.069444

0.930556

0.0694444*12-0.930556*-1

-0.09722

8

8

0.097222

0.902778

0.0972222*8-0.902778*-1

-0.125

9

6

0.115741

0.884259

0.115741*6-0.884259*-1

-0.18981

10

6

0.125

0.875

0.125*6-0.875*-1

-0.125

11

6

0.125

0.875

0.125*6-0.875*-1

-0.125

12

6

0.115741

0.884259

0.115741*6-0.884259*-1

-0.18981

13

8

0.097222

0.902778

0.0972222*8-0.902778*-1

-0.125

14

12

0.069444

0.930556

0.0694444*12-0.930556*-1

-0.09722

15

17

0.046296

0.953704

0.0462963*17-0.953704*-1

-0.16667

16

30

0.027778

0.972222

0.0277778*30-0.972222*-1

-0.13889

17

60

0.013889

0.986111

0.0138889*60-0.986111*-1

-0.15278

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:

Dice

Number of Permutations

1,2,3

6

1,2,4

6

1,2,5

6

1,2,6

6

1,1,2

3

1,2,2

3

Total

30



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:

Dice

Number of Permutations

1,1,2

3

1,1,3

3

1,1,4

3

1,1,5

3

1,1,6

3

1,1,1

1

Total

16



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.

Dice

Number of Permutations

1,1,2

3

1,1,3

3

1,1,4

3

1,1,5

3

1,1,6

3

Total

15



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.

Dice

Number of Permutations

1,2,2

3

1,2,3

6

1,2,4

6

1,2,5

6

1,2,6

6

1,3,3

3

1,3,4

6

1,3,5

6

1,3,6

6

1,4,4

3

1,4,5

6

1,4,6

6

1,5,5

3

1,5,6

6

1,6,6

3

Total

75

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.

Event

Permutations

Probability

Pays

Return

Player rolls 3 ones

1

0.00463

3

0.013889

Player rolls 2 ones

15

0.069444

2

0.138889

Player roll 1 one

75

0.347222

1

0.347222

Player rolls 0 ones

125

0.578704

-1

-0.5787

Total

216

1

-0.0787

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

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