Editing Liar's dice (section)

===Dice odds===
For a given number of unknown dice ''n'', the probability that ''exactly'' a certain quantity ''q'' of any face value are showing, ''P(q)'', is

:<math>\ P(q) = C(n,q) \cdot (1/6)^q \cdot (5/6)^{n-q}</math>

Where ''C(n,q)'' is the [[binomial coefficient|number of unique subsets]] of ''q'' dice out of the set of ''n'' unknown dice. In other words, the number of dice with any particular face value follows the [[binomial distribution]] <math>B(n,\tfrac{1}{6})</math>.

For the same n, the probability ''P'(q)'' that ''at least q'' dice are showing a given face is the sum of ''P(x)'' for all ''x'' such that ''q ≤ x ≤ n'', or

:<math>\ P'(q) = \sum_{x=q}^n C(n,x) \cdot (1/6)^x \cdot (5/6)^{n-x}</math>

These equations can be used to calculate and chart the probability of exactly ''q'' and at least ''q'' for any or multiple ''n''. For most purposes, it is sufficient to know the following facts of dice probability:

* The expected quantity of any face value among a number of unknown dice is one-sixth the total unknown dice.
* A bid of the expected quantity (or twice the expected value when playing with wilds), rounded down, has a greater than 50% chance of being correct and the highest chance of being exactly correct.<ref name="Christopher P Ferguson">{{cite web|last2=Ferguson|first2=Thomas S|author2-link= Thomas S. Ferguson |title=Models for the Game of Liar's Dice|url=https://www.math.ucla.edu/~tom/papers/LiarsDice.pdf|publisher=University of California at Los Angeles|access-date=16 January 2013|last1=Ferguson|first1=Christopher P|author1-link=Chris Ferguson}}</ref>