# 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 Probability 2 3 4 5 6 7 8 9 10 11 12 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.

### Other articles related to "probability":

Triangular Distribution
... In probability theory and statistics, the triangular distribution is a continuous probability distribution with lower limit a, upper limit b and mode c, where a < b and a ≤ c ≤ b ... The probability density function is given by whose cases avoid division by zero if c = a or c = b ...
Marilyn Vos Savant - Errors in The Column
... Obviously, the probability of an employee being chosen in one quarter is 25 percent ... Marilyn's response was The probability remains 25 percent, despite the repeated testing ... the size of the pool remains the same, so does the probability ...
Confidence Interval - Statistical Theory - Definition
... Let X be a random sample from a probability distribution with statistical parameters θ, which is a quantity to be estimated, and φ, representing quantities that are not of immediate ... Here Prθ,φ indicates the probability distribution of X characterised by (θ, φ) ... (u(X), v(X)) covers the unknown value θ with a high probability no matter what the true value of θ actually is ...
Probability - Relation To Randomness
... universe, based on Newtonian concepts, there would be no probability if all conditions are known, (Laplace's demon) ... Probability theory is required to describe quantum phenomena ...

### 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)