# Games 1: 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