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 ≤ ks 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) \
& = 5cdotfrac{1}{6}cdotfrac{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.

Read more about this topic:  Dice Cup

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