**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 *F*_{s,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,

Equivalently, the probability can be calculated using combinations:

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

Read more about this topic: Dice Cup

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### Famous quotes containing the word probability:

“Legends of prediction are common throughout the whole Household of Man. Gods speak, spirits speak, computers speak. Oracular ambiguity or statistical *probability* provides loopholes, and discrepancies are expunged by Faith.”

—Ursula K. Le Guin (b. 1929)

“Only in Britain could it be thought a defect to be “too clever by half.” The *probability* is that too many people are too stupid by three-quarters.”

—John Major (b. 1943)

“Liberty is a blessing so inestimable, that, wherever there appears any *probability* of recovering it, a nation may willingly run many hazards, and ought not even to repine at the greatest effusion of blood or dissipation of treasure.”

—David Hume (1711–1776)