Dice

What is a die?

For our purposes, a die is any element of S(6).

Although the labels ("♣","♦","♥","♠","♔","⚓") are sometimes used, we usually retain use of the labels ("1",...,"6").

The die (1/6,...,1/6) is called the fair die.

A physical die used for the purposes of gaming will be called an actual die. From the mathematical point of view, the set of actual dice is represented by a small-diameter set of dice. The meaning of small is a matter for experimental investigation. Likewise, the centre of the set is also a matter for experimentation but, in the absence of such experiments, is traditionally taken as being the fair die.

What is a roll of a die?

A 'roll' of the die f is defined algorithmically.

We use the computer's RANDOM function to select a number in ]0,1].

• If that number is in ]0,f(1)] we say that we have 'rolled "1" ', or that the die has 'come down "1" ';
• if it is in ]f(1),f(1)+f(2)], that we have 'rolled "2" ', or that the die has 'come down "2" ';
• if it is in ]f(1)+f(2),f(1)+f(2)+f(3)] that we have 'rolled "3" ', or that the die has 'come down "3" ';
• ...;
• if it is in ]f(1)+...+f(5),f(1)+...+f(6)], =]1-f(6),1], that we have 'rolled "6" ', .or that the die has 'come down "6" ';

The probability of "i", given f, is the likeliness over the singleton set {f} of "i". This is f(i), so the probability of "i" is f(i).