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# Sic Bo Appendix

## Introduction

Image take from the Claridge Hotel/Casino rule book. Click on the image to see the rules on the next page.

^{3}=216 possible combinations of the throw ofthree dice.

**Low**:This bet would have nohouse edge if it were not for the triple exception. Theprobability of a triple 1, 2, or 3 is 3/216. Theprobability of any total between 3 and 10 is 1/2, or108/216. So the probability of a winning triple is108/216 - 3/216 = 105/216. The player's edge is thus(105/216)*(+1) + (111/216)*(-1) = -6/216 =~-2.78%.

**High**: See Low.**SpecificNumber**:The probability ofrolling zero of a specific number is (5/6)

^{3}=125/216.

The probability of rolling one of a specific number is3*(1/6)^{1}*(5/6)^{2}= 75/216.

The probability of rolling two of a specific number is3*(1/6)^{2}*(5/6)^{1}= 15/216.

The probability of rolling three of a specific number is(1/6)^{3}= 1/216.

The player's edge is thus (125/216)*(-1) + (75/216)*(+1)+ (15/216)*(+2) + (1/216)*(+3) = -17/216 =~-7.780%.**Total of4**:There are 3 ways to rolla 4: (1+1+2, 1+2+1, 2+1+1). The player's edge is thus3/216*(+60) + (213/216)*(-1) = -33/216 =-15.278%.

**Total of5**:There are 6 ways to rolla 5: (1+1+3, 1+3+1, 3+1+1, 1+2+2, 2+1+2, 2+2+1). Theplayer's edge is thus 6/216*(+30) + (210/216)*(-1) =-30/216 = -13.889%.

**Total of6**:There are 10 ways to rolla 6: (1+1+4, 1+4+1, 4+1+1, 1+2+3, 1+3+2, 2+1+3, 2+3+1,3+1+2, 3+2+1, 2+2+2). The player's edge is thus10/216*(+17) + (206/216)*(-1) = -36/216 =-16.667%.

**Total of7**:There are 15 ways to rolla 7: (1+1+5, 1+5+1, 5+1+1, 1+2+4, 1+4+2, 2+1+4, 2+4+1,4+1+2, 4+2+1, 1+3+3, 3+1+3, 3+3+1, 2+2+3, 2+3+2, 3+2+2).The player's edge is thus 15/216*(+12) + (201/216)*(-1) =-21/216 = -9.722%.

**Total of8**:There are 21 ways to rolla 8: (1-6-6 * 3 ways, 1-2-5 * 6 ways, 1-3-4 * 6 ways,2-2-4 * 3 ways, 2-3-3 * 3 ways). The player's edge isthus 21/216*(+8) + (195/216)*(-1) = -27/216 =-12.500%.

**Total of9**:There are 25 ways to rolla 9: (1-2-6 * 6 ways, 1-3-5 * 6 ways, 1-4-4 * 3 ways,2-2-5 * 3 ways, 2-3-4 * 6 ways, 3-3-3 * 1 way). Theplayer's edge is thus 25/216*(+6) + (191/216)*(-1) =-41/216 = -18.982%.

**Total of10**:There are 27 ways to rolla 10: (1-3-6 * 6 ways, 1-4-5 * 6 ways, 2-2-6 * 3 ways,2-3-5 * 6 ways, 2-4-4 * 3 ways, 3-3-4 * 3 ways). Theplayer's edge is thus 27/216*(+6) + (189/216)*(-1) =-27/216 = -12.500%.

**Total of 11**: Seetotal of 10**Total of 12**: Seetotal of 9**Total of 13**: Seetotal of 8**Total of 14**: Seetotal of 7**Total of 15**: Seetotal of 6**Total of 16**: Seetotal of 5**Total of 17**: Seetotal of 4**Twonumbers**:Lets suppose the twonumbers chosen are 1 and 2. There are 30 combinationsfeaturing a 1 and a 2: 1-2-1 * 3 ways, 1-2-2 * 3 ways,1-2-3 * 6 ways, 1-2-4 * 6 ways, 1-2-5 * 6 ways, 1-2-6 * 6ways. The player's edge is thus 30/216*(+5) +(186/216)*(-1) = -36/216 = -16.667%.

**Specifictriplet**:There is only 1 way toroll a specific triplet. The player's edge is thus1/216*(+180) + (215/216)*(-1) = -35/216 =-16.20%.

**Anytriplet**:There are 6 ways to rolla triplet. The player's edge is thus 6/216*(+30) +(210/216)*(-1) = -30/216 = -13.889%.

**Specificpair**:Lets assume the pair chosen is ones. There are 16 ways two or three of thatnumber can be rolled: 1+1+1, 1+1+2 * 3 ways, 1+1+3 * 3 ways, 1+1+4 * 3 ways, 1+1+5 * 3 ways, 1+1+6 * 3 ways. Theplayer's edge is thus 16/216*(+10) + (200/216)*(-1) = -72/216 = -18.52%.

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} * [-1^{0}*combin(3,0)*combin(11-6*0-1,3-1) + -1^{1}*combin(3,1)*combin(11-6*1-1,3-1) ] =

1/216 * [1*1*combin(10,2) + -1*3*combin(4,2)] =

1/216 * [1*1*45 + -1*3*6] =

1/216 * [45-18] = 27/216 = 12.5%

Written by:Michael Shackleford