Sic-Bo and Dice – An Analysis

Sic-Bo is really popular in Asia.  This game involves 3 dice in a tumbler or variant which is tumbled, with varying payments for specific combinations of the dice.

There is an amazing way to calculate how many possible outcomes for a particular score from 3 six-sided dice there can be.  This involves the use of polynomial multiplication.

Here’s a simple way to look at this…

You have 3 dice with numbers from 1 to 6 on each of them.

Die 1	1	2	3	4	5	6
Die 2	1	2	3	4	5	6
Die 3	1	2	3	4	5	6

Now, add each number from Die 1 to the first number of Die 2 and then the second and so on like this.

	Die 1
Die 2	1	2	3	4	5	6
1	2	3	4	5	6	7
2	3	4	5	6	7	8
3	4	5	6	7	8	9
4	5	6	7	8	9	10
5	6	7	8	9	10	11
6	7	8	9	10	11	12

Now, add each of these 36 results to the first number of Die 3 and then the second and so on like this.

	Die 3
Die 1 and 2 results	1	2	3	4	5	6
2	3	4	5	6	7	8
3	4	5	6	7	8	9
4	5	6	7	8	9	10
5	6	7	8	9	10	11
6	7	8	9	10	11	12
7	8	9	10	11	12	13
3	4	5	6	7	8	9
4	5	6	7	8	9	10
5	6	7	8	9	10	11
6	7	8	9	10	11	12
7	8	9	10	11	12	13
8	9	10	11	12	13	14
4	5	6	7	8	9	10
5	6	7	8	9	10	11
6	7	8	9	10	11	12
7	8	9	10	11	12	13
8	9	10	11	12	13	14
9	10	11	12	13	14	15
5	6	7	8	9	10	11
6	7	8	9	10	11	12
7	8	9	10	11	12	13
8	9	10	11	12	13	14
9	10	11	12	13	14	15
10	11	12	13	14	15	16
6	7	8	9	10	11	12
7	8	9	10	11	12	13
8	9	10	11	12	13	14
9	10	11	12	13	14	15
10	11	12	13	14	15	16
11	12	13	14	15	16	17
7	8	9	10	11	12	13
8	9	10	11	12	13	14
9	10	11	12	13	14	15
10	11	12	13	14	15	16
11	12	13	14	15	16	17
12	13	14	15	16	17	18

Now, by counting the number of instances that each number appears in this table, we would have the number of possible outcomes for each score.  This works on any number of dice.

Points	Number of possible outcomes	Probability (Win)	Probability (Loss)	Expectation	Break-Even Odds
3	1	0.0046	0.9954	-0.9907	215.00
4	3	0.0139	0.9861	-0.9722	71.00
5	6	0.0278	0.9722	-0.9444	35.00
6	10	0.0463	0.9537	-0.9074	20.60
7	15	0.0694	0.9306	-0.8611	13.40
8	21	0.0972	0.9028	-0.8056	9.29
9	25	0.1157	0.8843	-0.7685	7.64
10	27	0.1250	0.8750	-0.7500	7.00
11	27	0.1250	0.8750	-0.7500	7.00
12	25	0.1157	0.8843	-0.7685	7.64
13	21	0.0972	0.9028	-0.8056	9.29
14	15	0.0694	0.9306	-0.8611	13.40
15	10	0.0463	0.9537	-0.9074	20.60
16	6	0.0278	0.9722	-0.9444	35.00
17	3	0.0139	0.9861	-0.9722	71.00
18	1	0.0046	0.9954	-0.9907	215.00

The following table lists the most common wager types and their respective details.

Wager	Number of possible outcomes	Probability (Win)	Probability (Loss)	Expectation	Break-Even Odds
Specific Triple	6 combinations and 1 possible outcome each	0.0046	0.9954	-0.9907	215
Any Triple	6	0.0278	0.9722	-0.9444	35.00
Specific Doubles	16 (15 + 1(the specific triple of that number))	0.0741	0.9259	-0.8519	12.50
Big/Small	105 (108 possible outcomes – 111,222,333 (for small) ,444,555 and 666 (for big))	0.4861	0.5139	-0.0278	1.06
Even/Odd	105 (108 possible outcomes – 111,333,555 (for odd) ,222,444 and 666 (for even))	0.4861	0.5139	-0.0278	1.06
2 Dice and Single Die Combination	30 combinations and 3 possible outcomes for each	0.0138	0.9861	-0.9722	71
3 Single Die Combination	20 combinations and 6 possible outcomes for each	0.0277	0.9722	-0.9444	35

Points refers to the total value from a 3-dice roll, from 3 to 18.

The number of possible outcomes is derived from tabulating all possible outcomes from a 3-dice roll. The total number of possible outcomes is 216 or 6 x 6 x 6 (this is the formula for a permutation where repetition is allowed).

Probability (Win) refers to the possible outcomes for each number divided by the total possible outcomes from a 3-dice roll.

Example:

For point 5, 6/216 = 0.0278

For point 10, 27/216 = 0.1250

Probability (Loss) is 1-Probability (Win).

Example:

For point 5, 1-0.0278=0.9722

For point 9, 1-0.1157=0.8843

Expectation is Probability (Win) – Probability (Loss).

Example:

For point 9, 0.1157 – 0.8843 = -0.7685

In the table the expectation has a ‘-‘ in front as this is in the perspective of the player having a negative expectation; which is positive for the house.

Break-even odds refers to the payment odds at which expectation is 0.  This means that the house should ALWAYS pay BELOW these odds in order to maintain a positive expectation for the house.

The is derived from the Probability (Loss)/ Probability (Win),

Example:

For point 3,  0.9954/0.0046 = 215

For point 6, 0.9537/0.0463 = 20.6

After computing all the possible outcomes of a 3-dice roll, we can see that the possible outcomes and probability follow the central limit theorem nicely.  The expectation dovetails the probability, being the result of 1-probability.

Single Die

For Single Die wagers, the math is different as a single wager wins when the specific number appears 1, 2 or 3 times. So, what we have to do is consider all the probabilities together.

Wager	Permutations	Probability	Break Even	Exp.
1 Die	75	0.3472	0.56	0.19292
2 Dice	15	0.0694	2.78	0.19292
3 Dice	1	0.0046	41.67	0.19292
Loss	125	0.5788	-1	-0.5788
			Total Exp.	0

The method of getting the break even is the most straight-forward.

Divide the Probability of a Loss by the total number of winning wager types as below:

0.5788 / 3 = 0.19292

Then divide this by the probability of the individual winning wager:

Wager	Working	Probability	Break Even
1 Die	0.19292 / 0.3472	0.3472	0.5556551
2 Dice	0.19292 / 0.0694	0.0694	2.7798769
3 Dice	0.19292 / 0.00462963	0.00462963	41.671467

You will find that by multiplying each wagers probability by the break even, you would have the same number, except positive, as that of the probability of losing. Adding the 2 numbers together would give you an expectation of 0.

This is similar to slot mathematics, which we might get into later at some point.

Number of possible outcomes for each point value from 3 to 18

Probability for each point value from 3 to 18

Expectation for each point value from 3 to 18

The following table shows a similar tabulation for 3-Single Dice combinations and Double and Single Dice combinations.  Again, the total possible number of outcomes is 216, with the number of categories being 56:

S/No	Combination	Possible Outcomes	Probability (Win)	Probability (Loss)	Expectation	Break-even odds
1	666	1	0.0046	0.9954	-0.9907	215
2	333	1	0.0046	0.9954	-0.9907	215
3	111	1	0.0046	0.9954	-0.9907	215
4	444	1	0.0046	0.9954	-0.9907	215
5	222	1	0.0046	0.9954	-0.9907	215
6	555	1	0.0046	0.9954	-0.9907	215
7	244	3	0.0139	0.9861	-0.9722	71
8	334	3	0.0139	0.9861	-0.9722	71
9	144	3	0.0139	0.9861	-0.9722	71
10	335	3	0.0139	0.9861	-0.9722	71
11	166	3	0.0139	0.9861	-0.9722	71
12	336	3	0.0139	0.9861	-0.9722	71
13	223	3	0.0139	0.9861	-0.9722	71
14	344	3	0.0139	0.9861	-0.9722	71
15	225	3	0.0139	0.9861	-0.9722	71
16	355	3	0.0139	0.9861	-0.9722	71
17	233	3	0.0139	0.9861	-0.9722	71
18	366	3	0.0139	0.9861	-0.9722	71
19	114	3	0.0139	0.9861	-0.9722	71
20	116	3	0.0139	0.9861	-0.9722	71
21	266	3	0.0139	0.9861	-0.9722	71
22	445	3	0.0139	0.9861	-0.9722	71
23	155	3	0.0139	0.9861	-0.9722	71
24	446	3	0.0139	0.9861	-0.9722	71
25	224	3	0.0139	0.9861	-0.9722	71
26	455	3	0.0139	0.9861	-0.9722	71
27	113	3	0.0139	0.9861	-0.9722	71
28	466	3	0.0139	0.9861	-0.9722	71
29	115	3	0.0139	0.9861	-0.9722	71
30	122	3	0.0139	0.9861	-0.9722	71
31	226	3	0.0139	0.9861	-0.9722	71
32	556	3	0.0139	0.9861	-0.9722	71
33	112	3	0.0139	0.9861	-0.9722	71
34	566	3	0.0139	0.9861	-0.9722	71
35	255	3	0.0139	0.9861	-0.9722	71
36	133	3	0.0139	0.9861	-0.9722	71
37	235	6	0.0278	0.9722	-0.9444	35
38	236	6	0.0278	0.9722	-0.9444	35
39	135	6	0.0278	0.9722	-0.9444	35
40	125	6	0.0278	0.9722	-0.9444	35
41	124	6	0.0278	0.9722	-0.9444	35
42	245	6	0.0278	0.9722	-0.9444	35
43	346	6	0.0278	0.9722	-0.9444	35
44	456	6	0.0278	0.9722	-0.9444	35
45	356	6	0.0278	0.9722	-0.9444	35
46	246	6	0.0278	0.9722	-0.9444	35
47	146	6	0.0278	0.9722	-0.9444	35
48	156	6	0.0278	0.9722	-0.9444	35
49	345	6	0.0278	0.9722	-0.9444	35
50	256	6	0.0278	0.9722	-0.9444	35
51	234	6	0.0278	0.9722	-0.9444	35
52	126	6	0.0278	0.9722	-0.9444	35
53	145	6	0.0278	0.9722	-0.9444	35
54	123	6	0.0278	0.9722	-0.9444	35
55	136	6	0.0278	0.9722	-0.9444	35
56	134	6	0.0278	0.9722	-0.9444	35

So, how do you calculate the combinations and permutations mathematically?

Here’s a great method created by Dr. James Tanton, known as labelling – it’s effective and easy to learn!  (http://gdaymath.com/courses/permutations-and-combinations/)

Combinations

For combinations, we use 6 factorial or 6! as our numerator (due to the die being 6-sided). Our denominator would be a multiplication of the numbers involved in the combination and those not, with the total being 6 as well. Here’re some examples:

Specific Triples: 6! / (1! X 5!) = 720 / (1 x 120) = 6 combinations. (Notice that 6! / (1! X 5!) has the numerator equalling 6 and the denominator also equalling 6 – that’s how you check the maths!)

Remember that there are 6 numbers, so we have 6/6 = 1 combination for each number. This would apply in the next example too.

Specific Doubles: 6! / (1! X 1! X 4!) = 720 / (1 x 1 x 24) = 30 combinations (Notice how the numerator and denominator also equal 6? Notice the denominator 1! X 1! X 4!, as 1 of the numbers would be a double, 1 of the numbers would be a single and the other 4 numbers would not be in the combination.)

Remember that for Specific Doubles there are 6 numbers, so each number would have 30 / 6 = 5 combinations for each number.

3 Single Die Combination: 6!/(3! X 3!) = 720 / (6 x 6) = 720 / 36 = 20 combinations. (We have 3! as 3 of the numbers of the 6 are singles included in the combination, with the other 3 numbers out of the combination.)

Total Combinations: (3 + 6 – 1)! / (3! x (6 – 1)! = 8!/(3! x 5!) = 40,320 / 720 = 56 total combinations

Permutations

For permutations, we use 3 factorial or 3! as our numerator (due to there being 3 dice in the game).

Specific Triples: 3!/3! = 6/6 = 1 permutation per combination (our denominator is 3! as all 3 numbers would have to be the same in order to make a triple.)

Specific Doubles: 3!/2! X 1! = 6/2 = 3 permutations per combination (our denominator is 2! X 1! as 2 of the numbers must be the same to make a double, with the last being any other number.)

3 Single Die Combination: 3!/(1! X 1! X 1!) = 6 permutations per combination (our denominator is 1! X 1! X 1! as all 3 numbers have to be difference to make a 3 die combination.)

Total permutations: 6 x 6 x 6 = 216 total permutations

Appendix: Here is a table listing all the possible outcomes on Sic-Bo.

D1	D2	D3	Points	Combination
1	1	1	3	111
2	1	1	4	112
3	1	1	5	113
4	1	1	6	114
5	1	1	7	115
6	1	1	8	116
1	2	1	4	112
2	2	1	5	122
3	2	1	6	123
4	2	1	7	124
5	2	1	8	125
6	2	1	9	126
1	3	1	5	113
2	3	1	6	123
3	3	1	7	133
4	3	1	8	134
5	3	1	9	135
6	3	1	10	136
1	4	1	6	114
2	4	1	7	124
3	4	1	8	134
4	4	1	9	144
5	4	1	10	145
6	4	1	11	146
1	5	1	7	115
2	5	1	8	125
3	5	1	9	135
4	5	1	10	145
5	5	1	11	155
6	5	1	12	156
1	6	1	8	116
2	6	1	9	126
3	6	1	10	136
4	6	1	11	146
5	6	1	12	156
6	6	1	13	166
1	1	2	4	112
2	1	2	5	122
3	1	2	6	123
4	1	2	7	124
5	1	2	8	125
6	1	2	9	126
1	2	2	5	122
2	2	2	6	222
3	2	2	7	223
4	2	2	8	224
5	2	2	9	225
6	2	2	10	226
1	3	2	6	123
2	3	2	7	223
3	3	2	8	233
4	3	2	9	234
5	3	2	10	235
6	3	2	11	236
1	4	2	7	124
2	4	2	8	224
3	4	2	9	234
4	4	2	10	244
5	4	2	11	245
6	4	2	12	246
1	5	2	8	125
2	5	2	9	225
3	5	2	10	235
4	5	2	11	245
5	5	2	12	255
6	5	2	13	256
1	6	2	9	126
2	6	2	10	226
3	6	2	11	236
4	6	2	12	246
5	6	2	13	256
6	6	2	14	266
1	1	3	5	113
2	1	3	6	123
3	1	3	7	133
4	1	3	8	134
5	1	3	9	135
6	1	3	10	136
1	2	3	6	123
2	2	3	7	223
3	2	3	8	233
4	2	3	9	234
5	2	3	10	235
6	2	3	11	236
1	3	3	7	133
2	3	3	8	233
3	3	3	9	333
4	3	3	10	334
5	3	3	11	335
6	3	3	12	336
1	4	3	8	134
2	4	3	9	234
3	4	3	10	334
4	4	3	11	344
5	4	3	12	345
6	4	3	13	346
1	5	3	9	135
2	5	3	10	235
3	5	3	11	335
4	5	3	12	345
5	5	3	13	355
6	5	3	14	356
1	6	3	10	136
2	6	3	11	236
3	6	3	12	336
4	6	3	13	346
5	6	3	14	356
6	6	3	15	366
1	1	4	6	114
2	1	4	7	124
3	1	4	8	134
4	1	4	9	144
5	1	4	10	145
6	1	4	11	146
1	2	4	7	124
2	2	4	8	224
3	2	4	9	234
4	2	4	10	244
5	2	4	11	245
6	2	4	12	246
1	3	4	8	134
2	3	4	9	234
3	3	4	10	334
4	3	4	11	344
5	3	4	12	345
6	3	4	13	346
1	4	4	9	144
2	4	4	10	244
3	4	4	11	344
4	4	4	12	444
5	4	4	13	445
6	4	4	14	446
1	5	4	10	145
2	5	4	11	245
3	5	4	12	345
4	5	4	13	445
5	5	4	14	455
6	5	4	15	456
1	6	4	11	146
2	6	4	12	246
3	6	4	13	346
4	6	4	14	446
5	6	4	15	456
6	6	4	16	466
1	1	5	7	115
2	1	5	8	125
3	1	5	9	135
4	1	5	10	145
5	1	5	11	155
6	1	5	12	156
1	2	5	8	125
2	2	5	9	225
3	2	5	10	235
4	2	5	11	245
5	2	5	12	255
6	2	5	13	256
1	3	5	9	135
2	3	5	10	235
3	3	5	11	335
4	3	5	12	345
5	3	5	13	355
6	3	5	14	356
1	4	5	10	145
2	4	5	11	245
3	4	5	12	345
4	4	5	13	445
5	4	5	14	455
6	4	5	15	456
1	5	5	11	155
2	5	5	12	255
3	5	5	13	355
4	5	5	14	455
5	5	5	15	555
6	5	5	16	556
1	6	5	12	156
2	6	5	13	256
3	6	5	14	356
4	6	5	15	456
5	6	5	16	556
6	6	5	17	566
1	1	6	8	116
2	1	6	9	126
3	1	6	10	136
4	1	6	11	146
5	1	6	12	156
6	1	6	13	166
1	2	6	9	126
2	2	6	10	226
3	2	6	11	236
4	2	6	12	246
5	2	6	13	256
6	2	6	14	266
1	3	6	10	136
2	3	6	11	236
3	3	6	12	336
4	3	6	13	346
5	3	6	14	356
6	3	6	15	366
1	4	6	11	146
2	4	6	12	246
3	4	6	13	346
4	4	6	14	446
5	4	6	15	456
6	4	6	16	466
1	5	6	12	156
2	5	6	13	256
3	5	6	14	356
4	5	6	15	456
5	5	6	16	556
6	5	6	17	566
1	6	6	13	166
2	6	6	14	266
3	6	6	15	366
4	6	6	16	466
5	6	6	17	566
6	6	6	18	666