SicBo 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 sixsided 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 
BreakEven 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 
BreakEven 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 3dice roll, from 3 to 18.
The number of possible outcomes is derived from tabulating all possible outcomes from a 3dice 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 3dice roll.
Example:
For point 5, 6/216 = 0.0278
For point 10, 27/216 = 0.1250
Probability (Loss) is 1Probability (Win).
Example:
For point 5, 10.0278=0.9722
For point 9, 10.1157=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.
Breakeven 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 3dice 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 1probability.
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 3Single 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 
Breakeven 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/permutationsandcombinations/)
Combinations
For combinations, we use 6 factorial or 6! as our numerator (due to the die being 6sided). 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 SicBo.
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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