# Normalisation (probability)

In mathematical probability equations, which are used in nearly all branches of science, a **normalization** constant (or function) is often used to ensure that the sum of all probabilities totals one, or

Probability distributions can be divided into two main groups: discrete probability distributions and continuous probability distributions.

## Discrete Probability Distributions

Discrete probability distributions are used throughout gaming theory. Consider the simple example of rolling a pair of six-sided dice. Summing up the total roll of the dice yields the following possibilities:

Total (i) | Possible outcomes (Die1,Die2) | Occurrences (n_{i}) |
---|---|---|

2 | (1,1) | 1 |

3 | (1,2), (2,1) | 2 |

4 | (1,3), (3,1), (2,2) | 3 |

5 | (1,4), (4,1), (2,3), (3,2) | 4 |

6 | (1,5), (5,1), (2,4), (4,2), (3,3) | 5 |

7 | (1,6), (6,1), (2,5), (5,2), (3,4), (4,3) | 6 |

8 | (2,6), (6,2), (5,3), (3,5), (4,4) | 5 |

9 | (3,6), (6,3), (4,5), (5,4) | 4 |

10 | (4,6), (6,4), (5,5) | 3 |

11 | (5,6), (6,5) | 2 |

12 | (6,6) | 1 |

Since the probability of any particular outcome is proportional to the number of ways it can occur

where is a coefficient of probability for outcome i. Assuming the dice are symmetrical we assume all values of are equal and their sum equals 1.

Solving for N yields 1/36, the number of possible outcomes, so that the probability of total = i occurring are

, and the sum of all probabilities is one

## Continuous probability distributions

In most scientific equations, probability functions are continuous functions, and the probability coefficients are sometimes functions rather than constants. For example, the zeta distribution with parameter *s* assigns probability proportional to 1/*n*^{s} to the integer *n*: the normalizing factor is then the value of the Riemann zeta function