Craps Side Bets Odds Calculation

2	3	4	5	6
2	3	4	6	5
2	3	5	4	6
2	3	5	6	4
2	3	6	4	5
2	3	6	5	4
2	4	3	5	6
2	4	3	6	5
2	4	5	3	6
2	4	5	6	3
2	4	6	3	5
2	4	6	5	3
2	5	3	4	6
2	5	3	6	4
2	5	4	3	6
2	5	4	6	3
2	5	6	3	4
2	5	6	4	3
2	6	3	4	5
2	6	3	5	4
2	6	4	3	5
2	6	4	5	3
2	6	5	3	4
2	6	5	4	3
3	2	4	5	6
3	2	4	6	5
3	2	5	4	6
3	2	5	6	4
3	2	6	4	5
3	2	6	5	4
3	4	2	5	6
3	4	2	6	5
3	4	5	2	6
3	4	5	6	2
3	4	6	2	5
3	4	6	5	2
3	5	2	4	6
3	5	2	6	4
3	5	4	2	6
3	5	4	6	2
3	5	6	2	4
3	5	6	4	2
3	6	2	4	5
3	6	2	5	4
3	6	4	2	5
3	6	4	5	2
3	6	5	2	4
3	6	5	4	2
4	2	3	5	6
4	2	3	6	5
4	2	5	3	6
4	2	5	6	3
4	2	6	3	5
4	2	6	5	3
4	3	2	5	6
4	3	2	6	5
4	3	5	2	6
4	3	5	6	2
4	3	6	2	5
4	3	6	5	2
4	5	2	3	6
4	5	2	6	3
4	5	3	2	6
4	5	3	6	2
4	5	6	2	3
4	5	6	3	2
4	6	2	3	5
4	6	2	5	3
4	6	3	2	5
4	6	3	5	2
4	6	5	2	3
4	6	5	3	2
5	2	3	4	6
5	2	3	6	4
5	2	4	3	6
5	2	4	6	3
5	2	6	3	4
5	2	6	4	3
5	3	2	4	6
5	3	2	6	4
5	3	4	2	6
5	3	4	6	2
5	3	6	2	4
5	3	6	4	2
5	4	2	3	6
5	4	2	6	3
5	4	3	2	6
5	4	3	6	2
5	4	6	2	3
5	4	6	3	2
5	6	2	3	4
5	6	2	4	3
5	6	3	2	4
5	6	3	4	2
5	6	4	2	3
5	6	4	3	2
6	2	3	4	5
6	2	3	5	4
6	2	4	3	5
6	2	4	5	3
6	2	5	3	4
6	2	5	4	3
6	3	2	4	5
6	3	2	5	4
6	3	4	2	5
6	3	4	5	2
6	3	5	2	4
6	3	5	4	2
6	4	2	3	5
6	4	2	5	3
6	4	3	2	5
6	4	3	5	2
6	4	5	2	3
6	4	5	3	2
6	5	2	3	4
6	5	2	4	3
6	5	3	2	4
6	5	3	4	2
6	5	4	2	3
6	5	4	3	2

Iterations=100000
Craps <- matrix(0,nrow=Iterations,ncol=11)
colnames(Craps) <- c(2:12)
Results <- matrix(0,nrow=2,ncol=3)
colnames(Results) <- c("Small","Tall","All")
rownames(Results) <- c("Frequency","Probability")

for (i in 1:Iterations)
{
  Roll=0
  while (Roll!=7)
  {
    Roll <- sum(sample(c(1:6),2,replace=TRUE))
    Craps[i,toString(Roll)] <- Craps[i,toString(Roll)]+1
  }
  #Small (rolls 2,3,4,5,6 before 7) 
  if (prod(Craps[i,1:5])>0)
    Results[1,"Small"]=Results[1,"Small"]+1
  #Tall (rolls 8,9,10,11,12 before 7)
  if (prod(Craps[i,7:11])>0)
    Results[1,"Tall"]=Results[1,"Tall"]+1
  #All (rolls 2,3,4,5,6,8,9,10,11,12 before 7)
  if (prod(Craps[i,1:11])>0)
    Results[1,"All"]=Results[1,"All"]+1
  i=i+1
}
Results[2,]=Results[1,]/Iterations
format(Results,scientific=FALSE)

                Small      Tall      All
Frequency   2.638e+03 2.564e+03 5.14e+02
Probability 2.638e-02 2.564e-02 5.14e-03

 
> ##################################################################
> # Odds against rolling a subset of numbers before a single number - The All bet
> ##################################################################
> nums = c(2,3,4,5,6,8,9,10,11,12,7)   # Last must occur only after all others in any order
> 
> in_36 = ifelse(nums <= 7, nums-1, 13-nums)  # Ways to make each number
> i = length(in_36)
> p = 0
> for (j in 1:(i-1)) {                # Last number before j numbers
+   terms = combn(in_36[1:(i-1)],j)   # Matrix w/combos of j numbers in C(i-1,j) columns
+   for (k in 1:ncol(terms)) {        # Sum each column, compute and add probabilities 
+     p = p + (-1)^(j+1) * in_36/(in_36 + sum(terms[1:j,k]))
+   }
+ }
> p1 = (1-p)
> p1
[1] 0.005257704
> ch = 1/(1-p)
> ch
[1] 190.1971
> Odds.to.1 = 1/(1-p) - 1
> Odds.to.1
[1] 189.1971


> ##################################################################
> # Odds against rolling a subset of numbers before a single number - Small or Tall
> ##################################################################
> nums = c(2,3,4,5,6,7)   # Last must occur only after all others in any order
> 
> in_36 = ifelse(nums <= 7, nums-1, 13-nums)  # Ways to make each number
> i = length(in_36)
> p = 0
> for (j in 1:(i-1)) {                # Last number before j numbers
+   terms = combn(in_36[1:(i-1)],j)   # Matrix w/combos of j numbers in C(i-1,j) columns
+   for (k in 1:ncol(terms)) {        # Sum each column, compute and add probabilities 
+     p = p + (-1)^(j+1) * in_36/(in_36 + sum(terms[1:j,k]))
+   }
+ }
> p1 = (1-p)
> p1
[1] 0.02635391
> ch = 1/(1-p)
> ch
[1] 37.94503
> Odds.to.1 = 1/(1-p) - 1
> Odds.to.1
[1] 36.94503 

#! /usr/bin/env python3
from fractions import Fraction

      #0  1  2  3  4  5  6  7  8  9 10 11 12
way = [0, 0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1]

memo = { }

def ways(numbers):
   w = 0;
   for r in numbers:
       w += way[r]
   return w

def p(these, those):
    if len(these) == 0: return 1;
    if len(these) == 1:
        for w in these:
            return Fraction(1, 1 + Fraction(ways(those), ways(these)))
    s = str(these) + str(those)
    if s in memo: return memo[s]
    v = Fraction(0)
    full = these | those
    for r in these:
        only_r = frozenset([r])
        full_but_r = full - only_r
        these_without_r = these - only_r
        v += p(only_r, full_but_r) * p(these_without_r, those)
    memo[s] = v
    return v

seven = frozenset([7])
small = frozenset([2, 3, 4, 5, 6])
tall  = frozenset([8, 9, 10, 11, 12])
full  = small | tall

print('small', p(small, seven))
print('tall ', p(tall,  seven))
print('full ', p(full,  seven))