Throwing Dice

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Possible outcomes of throwing a single die:

1. dots = Range[6]

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We define a uniform probability distribution for the results 1 to 6, and call it "dice":

1. dice = DiscreteUniformDistribution[{1, 6}];

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How many times we toss the die => trials:

1. trials = 120;

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Toss it:

1. toss = Table[RandomVariate[dice], trials]

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{3,3,6,5,2,3,2,5,3,2,1,2,3,6,4,5,1,4,6,4,3,5,4,6,1,2,4,2,4,2,1,2,3,6,5,2,1,1,1,3,4,2,1,4,2,5,1,5,1,4,2,4,2,5,1,2,6,6,2,6,5,4,5,2,6,2,4,3,2,1,4,4,5,5,2,3,3,5,6,2,3,5,6,4,5,4,2,3,2,4,5,1,2,4,2,3,2,1,4,4,2,3,1,1,5,6,1,1,2,4,5,2,1,4,2,2,1,6,2,5}

Count how many of each number (between 1/2 and 3/2, 3/2 and 5/2, etc):

1. freq = BinCounts[toss, {.5, 6.5, 1}]

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{20, 31, 15, 22, 19, 13}

Plot the frequency of each result as a histogram:

1. BarChart[freq]

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Efficient way to add up the elements of the list "dots" ==>  Apply the "Plus" operator to each element:

1. Plus @@ dots

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21

Calculate the average value of the number of dots:

1. N[Plus @@ (freq*dots) / trials]

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3.23333

Calculate the average value of the number of dots squared:

1. N[Plus @@ (freq*dots^2) / trials]

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13.1167

1. Rolldice[trials_] := (
2.   dots = Range[6];
3.   dice = DiscreteUniformDistribution[{1, 6}];
4.   toss = Table[RandomVariate[dice], trials];
5.   freq = BinCounts[toss, {.5, 6.5, 1}];
6.   BarChart[freq];
7.   avgdots = N[Plus @@ (freq*dots)/trials];
8.   avgdotssq = N[Plus @@ (freq*dots^2)/trials];
9.   sdev = Sqrt[avgdotssq - avgdots^2];
10.   Print["mean = ", avgdots, "  standard deviation = ", sdev])

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Let's define a function to evaluate mean and standard deviation:
Compare with the formulas for discrete uniform distribution:	\begin{aligned}\frac{6+1}2&=3.5\\\sqrt{\frac{6^2-1}{12}}&=1.70783\end{aligned}