Definition:Die

A die (plural: dice) is a device for generating random numbers, traditionally between 1 and 6.

They have been around for at least five thousand years, and have formed the basis of humanity's love-affair with gambling throughout history. Even the clothing of Jesus Christ was reportedly gambled for at his crucifixion in a dice game.

A die usually consists of a cube on which each face denotes a different number, usually (again) from 1 to 6.

The idea is that when you roll it, or throw it, or otherwise cause it to land upon a flattish surface, one of the sides will be face-up, and (with a good design of die) which side that is can not be precisely predicted.

Hence it serves as a generator of a discrete uniform distribution.

By using dice with different shapes (usually either platonic solids or prisms), a different probability space can be generated.

Coin-Tossing
At the limit of this process lies the art of coin-tossing or coin-flipping. This is a technique to select randomly one of two options. The usual scenario is to resolve an issue between two parties. The most usual of these is to determine which of the two parties is to choose whether to take the first move, or otherwise determine the starting arrangement, in a game or sport.

A coin is thrown into the air in such a manner that it turns over and over. This can be conveniently achieved by coiling the hand into a loose fist, balancing the coin on the first knuckle of the index finger, and then pinging the thumb up from under the index finger by a physical process derived from the science of catastrophe theory. The nail of the thumb will then impact smartly against the edge of the coin which projects over the edge of the index finger's knuckle, propelling it upwards in such a manner that, by an application of the physics of angular momentum, it will be caused to rotate more-or-less rapidly about a diameter.

When it descends, the side which will remain uppermost will be dependent upon both the speed of rotation and the distance travelled, neither of which is easy to gauge precisely enough to make the process predictable.

The descending coin will either be caught in the hand by the one flipping it (the "tosser"), to be immediately clasped to the back of the other hand, or allowed to fall directly to the ground. The other party (the "caller") is offered the chance of guessing which of the two sides of the coin is on top. The two sides are usually referred to (in the English-speaking world) as "heads" and "tails". The "heads" side tends to be the one that features the head of state of the nation to which the coin belongs, while the "tails" is the other side.

Once the caller has made the call, the uppermost side of the coin is revealed. If the side matches that called by the caller, the caller has won, and is given the option of taking the choice as to the starting configuration of the game. Otherwise the tosser has won, and takes the option.

Biased Coin
It is usual to consider that there is an equal probability for a coin to land either heads or tails. Such a coin is called a fair coin.

However, discs of metal can crafted to look like coins with a higher probability of landing on one side rather than the other.

Such a device is known as a biased coin.

It is worth mentioning that as the manufacture of coins is such that the design on either side is embossed in the metal itself, their physical nature is such that they are by definition asymmetric. Hence by their very nature all coins are to a certain extent biased to a more-or-less minute degree. Skilled tossers are rumoured to be able to take advantage of this fact.

Bernoulli Trial
The tossing of a coin can be modelled mathematically by means of a Bernoulli trial.