Introduction If you toss two fair six-sided dice, you will get a number between 2 and 12. Although each die is fair, the sum is not: the probability of getting a 2 is 1 36 , while the probability of a 7 is 16 . Thus the question arises: Can unfair (or loaded) dice lead to a fair sum? The answer is no, as was shown by Honsberger [4], using elementary methods. A proof using generating functions i...

Play is often episodic and mission-centric, with a series of challenges culminating in a final puzzle or enemy that must be overcome. Multiple missions played with the same characters may be related to each other in a plot arc of escalating challenges. The exact tone, structure, pace and end (if any) vary from game to game depending on the needs and preferences of the players, as in [9]. "THE C...

A throw of a fair die is commonly considered as a paradigm for chance. The die is usually a cube of a homogeneous material. The symmetry suggests that such a die has the same chance of landing on each of its six faces after a vigorous roll so it is considered to be fair. Generally, a die with a shape of convex polyhedron is fair by symmetry if and only if it is symmetric with respect to all its...

The probability of a high roll is larger for the biased dice than for the fair dice. Suppose that you draw a die at random from the box and roll it twice, observing a four on the first roll and a six on the second roll. What is the probability that the die is biased? A Bayesian analysis combines ones prior beliefs about the probability of a hypothesis with the likelihood. The likelihood is the ...

The probability of a high roll is larger for the biased dice than for the fair dice. Suppose that you draw a die at random from the box and roll it twice, observing a four on the first roll and a six on the second roll. What is the probability that the die is biased? A Bayesian analysis combines ones prior beliefs about the probability of a hypothesis with the likelihood. The likelihood is the ...

Given a random variable X which takes n equiprobable values, we consider several algorithmic questions related to the classical problem of simulating the outcomes of X by using a limited number of biased coins. in the framework of the project: \EEcienza di Algoritmi e Progetto di Strutture Informative".

'If God has made the world a perfect mechanism, He has at least conceded so much to our imperfect intellects that in order to predict little parts of it, we need not solve innumerable differential equations, but can use dice with fair success.'