In this post we will delve deeper into some probability concepts. In previous posts we talked about the concepts of functions, which according to Descartes is the act of saying what one hears. Given concepts of events and sample spaces, we will calculate the chances of these events occurring with the concept of a function, in this case probability.

Fundamentals

In previous posts, we talked about the concept of function. In this way, René Discards used an ear as a symbol.

Probability is a function that has as input the sample space $\Omega$ and output the probability of an event.

Relevance

For probabilities, we must have a notion of relevance.

  • $\in$ belongs
  • $\notin$ does not belong
  • $\subset$ contained
  • $\not\subset$ not contained
  • $\supset$ contains
  • $\not\supset$ does not contain

Event Class

Definition (Class of Events): Be it an experiment with sample space $\Omega,$, we define the class of events $\mathcal{A}$, which satisfy the properties:

  • $\varnothing \in \mathcal{A}$: empty belongs to the event class.
  • If $B\in \mathcal{A}$ then $\complement B \in \mathcal{A}$: a set B belongs to the class then the complement of B also belongs to the class.
  • If $D,B\in \mathcal{A}$ then $B\bigcup D \in \mathcal{A}:$ if two sets belong to the class, then the union belongs to the class.
$\varnothing \in \mathcal{A}$ $A\in \mathcal{A}$ then $\complement A \in \mathcal{A}$ $D,B\in \mathcal{A}$ then $B\bigcup D \in \mathcal{A}$

Probability Axioms

Definition (Probability): In an experiment with sample space $\Omega$ and event class $\mathcal{A}$, the probability is a function with input in the event class and output with values ​​between zero and 1.

The axioms that probability must be satisfied are:

  • $\mathbb{P}(\Omega)=1$ e $\mathbb{P}(\varnothing)=0$
  • For every event $A$ in the class, $\mathbb{P}(A)$ is between 0 and 1.
  • For any sequence of mutually exclusive events, for example, $B, C$ where $B\bigcap C = \varnothing$
$$\mathbb{P}(B\bigcup C)= \mathbb{P}(B) + \mathbb{P}(C)$$

So far are the main basic concepts of probability. Wait for the next posts.