A probability is calculated or measured. It defines the chance that an event has to occur.

Let's suppose that we have ten books: 7 of Mathematics and 3 of Physiscs, and we want to take one of them randomly. We assume that the events "take a book" from the set are equiprobable, that is all the outcomes have the same chance to be taken.

The event "take a Physics book" has three equiprobable results. Three favorable outcomes (results) can occur among ten possibles outcomes (results). the brobability to get a Physics book is then 3/10.

Generally, in a random experiment associated to a sample space containing N results mutually exlusive and equiprobable; if an event A has n_A favorable results, then the probability of the event A denoted by P(A) is n_A/N

The bprobability of the event A = number of favorable outcomes/number of possibles outcomes

This definition works when the sample space is finite and contains equiprobable results. When it is not the case, we use the experiment process. We repeat the random experiment a lot of times, let's N times. During all these N times, if we find n_A events A, then the probability of the event A is very close to n_A/N, whish is here the relative frequency of the event A. In the same conditions, the more the number of trials is large, the more the relative frequency is accurate and tends to a limite value which is the probability of the associated event. This principle is called Bernoulli theorem or law of the large numbers. This empirical definition of probability is then the limite of n_A/N when N tends to infinity (N very large).

Remark that N is said large when the relative frequency varies slowly when passing from N to N+1.

When the sample space is continue, that is not countable, we define probability by measurement. For example, when we throw a dart on a cercle of radius R , the probability to get the dart inside a circle of a radius r is the rate of the areas: πr²/πR² = r²/R²