Dice Cup - Probability

Probability

For a single roll of a fair s-sided die, the probability of rolling each value is exactly 1/s; this is an example of a discrete uniform distribution. For n multiple rolls, with a s-sided die the possibility space is equal to sn. So, for n rolls of an s-sided die the probability of any result is 1/sn.

However, if we are rolling two dice and adding the result together, as in the game craps, the total is distributed in a triangular curve; the case for common dice follows:

Sum 2 3 4 5 6 7 8 9 10 11 12
Probability 1/36
 2/36
=1/18		 3/36
=1/12		 4/36
=1/9		 5/36
 6/36
=1/6		 5/36
 4/36
=1/9		 3/36
=1/12		 2/36
=1/18		 1/36

As the number of dice increases, the distribution of the sum of all numbers tends to normal distribution by the central limit theorem; the exact value of a sum of n s-sided dice, k, is

where Fs,1(k) = 1/s for 1 ≤ k ≤ s and 0 otherwise.

A faster algorithm would adapt the exponentiation by squaring algorithm:

.

In the triangular curve described above,

\begin{align} F_{6,2}(6) & =\sum_n {F_{6,1}(n) F_{6,1}(6 - n)} \\ & =F_{6,1}(1) F_{6,1}(5) + F_{6,1}(2) F_{6,1}(4) + \\ & \qquad \cdots + F_{6,1}(5) F_{6,1}(1) \\ & = 5\cdot\frac{1}{6}\cdot\frac{1}{6}=\frac{5}{36}\approx0.14 \end{align}

Equivalently, the probability can be calculated using combinations:

where is the floor function. The probability of rolling an exact sequence of numbers is 1/sn.

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