DiscreteUniformDistribution
RandomVariate
Probability
ProductDistribution
PDF
Table

Probability & Statistics
Mathematics
Puzzles and Recreation

Dice Roll Probabilities in Craps

Explore how a dice model can be simulated and apply it to compute win and loss probabilities in craps

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A fair six-sided die can be modeled using a

DiscreteUniformDistribution

In[1]:=

dice=DiscreteUniformDistribution[{1,6}]

Out[1]=

DiscreteUniformDistribution[{1,6}]

Simulate 10 independent throws of the die:

In[2]:=

RandomVariate[dice,10]

Out[2]=

{1,1,3,6,6,3,2,2,6,2}

Model two fair dice as independent discrete uniform variables:

In[3]:=

craps=ProductDistribution[{DiscreteUniformDistribution[{1,6}],2}]

Out[3]=

ProductDistribution[{DiscreteUniformDistribution[{1,6}],2}]

The resulting PDF can be tabulated:

In[4]:=

crapsProb=Table[PDF[craps,{d1,d2}],{d1,6},{d2,6}];

In[5]:=

TableForm[crapsProb,TableHeadings{Range[6],Range[6]}]

Out[5]//TableForm=

	1	2	3	4	5	6
1	1 36	1 36	1 36	1 36	1 36	1 36
2	1 36	1 36	1 36	1 36	1 36	1 36
3	1 36	1 36	1 36	1 36	1 36	1 36
4	1 36	1 36	1 36	1 36	1 36	1 36
5	1 36	1 36	1 36	1 36	1 36	1 36
6	1 36	1 36	1 36	1 36	1 36	1 36

Find the probability of specific outcomes in the game. "Snake eyes":

In[6]:=

Probability[D1+D22,{D1,D2}craps]

Out[6]=

"Boxcars":

In[7]:=

Probability[D1+D212,{D1,D2}craps]

Out[7]=

"Eighter from Decatur":

In[8]:=

Probability[D1+D28,{D1,D2}craps]

Out[8]=

"Little Joe":

In[9]:=

Probability[D1+D24,{D1,D2}craps]

Out[9]=

Generate the full distribution for the sum of two dice:

In[10]:=

probs=Table[Probability[D1+D2k,{D1,D2}craps],{k,12}]

Out[10]=

0,



Plot these in a bar chart:

In[11]:=

BarChart[probs,ChartLabelsRange[1,12],PlotLabel"Probability of Dice Total"]

Out[11]=

Find the probability of losing in one throw or getting craps, i.e. any of the sums 2, 3, or 12:

In[12]:=

Probability[D1+D22∨D1+D23∨D1+D212,{D1,D2}craps]

Out[12]=

Find the probability of winning in one throw, i.e. getting the sums 7 or 11:

In[13]:=

Probability[D1+D27∨D1+D211,{D1,D2}craps]

Out[13]=

External Links

https://mathworld.wolfram.com/Craps.html

Related Symbols

DiscreteUniformDistribution
RandomVariate
Probability
ProductDistribution
PDF
Table

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Contributed by: Wolfram Staff