Probability of simple propositions

Let us start by recalling the basic rules of probability for propositions or hypotheses. Let $A$ and $B$ be propositions, which can take on only two values, for example, true or false. The notation $P(A)$ stands for the probability that $A$ is true. The elementary rules of probability for simple propositions are

$$0\le P(A) \le 1;$$ (1)

$$P(\Omega) = 1;$$ (2)

$$P(A\cup B) = P(A)+P(B) - P(A\cap B) \,.%%eq:basic3ext$$ (3)

$$P(A\cap B) = P(A\,\vert\,B) \, P(B) = P(B\,\vert\,A) \, P(A)\,,$$ (4)

where $\Omega$ means tautology (a proposition that is certainly true). The construct $A\cap B$ is true only when both $A$ and $B$ are true (logical AND), while $A\cup B$ is true when at least one of the two propositions is true (logical OR). $A\cap B$ is also written simply as `$A,B$' or $A B$, and is also called a logical product, while $A\cup B$ is also called a logical sum. $P(A, B)$ is called the joint probability of $A$ and $B$. $P(A\,\vert\,B)$ is the probability of $A$ under that condition that $B$ is true. We often read it simply as ``the probability of A, given $B$.'' .

Equation (4) shows that the joint probability of two events can be decomposed into conditional probabilities in different two ways. Either of these ways is called the product rule. If the status of $B$ does not change the probability of $A$, and the other way around, then $A$ and $B$ are said to be independent, probabilistically independent to be precise. In that case, $P(A\,\vert\,B)=P(A)$, and $P(B\,\vert\,A)=P(B)$, which, when inserted in Eq. (4), yields

$$P(A\cap B) = P(A) \, P(B)\hspace{.5cm} \Longleftrightarrow\hspace{.5cm}\mbox{\it probabilistic independence} \, .$$ (5)

Equations (1)-(4) logically lead to other rules which form the body of probability theory. For example, indicating the negation (or opposite) of $A$ with $\overline{A}$, clearly $A \cup \overline{A}$ is a tautology ( $A \cup \overline{A}=\Omega$), and $A \cap \overline{A}$ is a contradiction ( $A \cap \overline{A}=\emptyset$). The symbol $\emptyset$ stands for contradiction (a proposition that is certainly false). Hence, we obtain from Eqs. (2) and (3)

$$P(A) + P(\overline{A}) = 1\, ,$$ (6)

which says that proposition $A$ is either true or not true.