## Definitions

Probabilities are the branch of mathematics that calculates the probability of an event, that is, the frequency of an event in relation to all possible cases.
This branch of mathematics is born from the games of chance, more precisely the desire to predict the unpredictable or to quantify the uncertain. First of all it is necessary to specify what it is not: it does not make it possible to predict the result of a single experiment.
An experiment with results that one is able to describe, is said random when it verifies the condition:
-We do not know which of these results will occur when we perform the experiment.
An event is part of the set of possible outcomes
An elementary event is an event containing only one result
When performing a very large number of times a random experiment, the frequency of realization of an event approaches a ‘theoretical frequency’ called probability.

## Vocabulary

A probability is a number between 0 and 1. The sum of the probabilities of all the elementary events is equal to 1. An impossible event has probability 0. A certain event has for probability 1.
Two contrary events are events whose meeting is the certain event and the empty intersection. The sum of the probabilities of two contrary events is equal to 1.

## Properties

When all possible cases can be determined, the probability of an event is given by the quotient of the number of favorable cases by the number of possible cases.
For example, when playing face-to-face, the probability of getting face is because there is 1 favorable case to ‘face’ on 2 possible cases (‘falling on pile’ and ‘falling on face’).
The probability of the adverse event of event A is equal to 1-p (A). For example, when throwing a dice, the probability of getting a 4 is 1/6 and that of not getting 4 is 1-1/6= 5/6

## Some examples of cases

Calculating the probability of multiple events consists in breaking down the problem into separate probabilities. Here are three examples.

### 1st example:

What is the probability of getting two consecutive five by throwing a six-sided die?
The probability of getting a five is 1/6 and the probability of getting another five with the same die is also 1/6.
These are independent events. Indeed, the first throw does not affect the second throw. You can take out a 3, then another 3 immediately behind.

### Example 2:

Two cards are drawn at random from a deck of cards. What is the probability that both cards are shamrocks?
The probability that the first card is a clover is 13/52 or 1/4 (there are 13 clovers in each deck of cards). Now, the probability that the second card is a clover is 12/51.
Here, however, we are dealing with dependent events. What you do the first time influences what you do next. If you shoot a 3 of clubs and you do not put the card back in the game, you have a game with one less clover and one less card (51 instead of 52).