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.

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

Sic-Bo1

Probability for each point value from 3 to 18

Sic-Bo2

Expectation for each point value from 3 to 18

Sic-Bo3

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

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