Rules of probability

Figure: Venn diagrams and set properties.

The subjective definition of probability, together with the condition of coherence, requires that $0\le p\le 1$. This is one of the rules which probability has to obey. It is possible, in fact, to demonstrate that coherence yields to the standard rules of probability, generally known as axioms. At this point it is worth clarifying the relationship between the axiomatic approach and the others.

• Combinatorial and frequentistic ``definitions'' give useful rules for evaluating probability, although they do not, as it is often claimed, define the concept.
• In the axiomatic approach one refrains from defining what the probability is and how to evaluate it: probability is just any real number which satisfies the axioms. It is easy to demonstrate that the probabilities evaluated using the combinatorial and the frequentistic prescriptions do in fact satisfy the axioms.
• The subjective approach to probability, together with the coherence requirement, defines what probability is and provides the rules which its evaluation must obey; these rules turn out to be the same as the axioms.

Since everybody is familiar with the axioms and with the analogy $events\Leftrightarrow sets$ (see Table  and Fig. ) let us remind ourselves of the rules of probability in this form:

Table: Events versus sets.

Events	Sets
		Symbol
event	set	$E$
certain event	sample space	$\Omega$
impossible event	empty set	$\emptyset$
implication	inclusion	$E_1\subseteq E_2$
	(subset)
opposite event	complementary set	$\overline{E}$ ( $E\cup \overline{E} = \Omega$)
(complementary)
logical product (``AND'')	intersection	$E_1 \cap E_2$
logical sum (``OR'')	union	$E_1 \cup E_2$
incompatible events	disjoint sets	$E_1 \cap E_2 = \emptyset$
complete class	finite partition	$\left\{ \begin{array}{l} E_i \cap E_j = \emptyset \ \ \forall\, i\ne j\\ \cup_i E_i = \Omega \end{array}\right.$

Axiom 1

$0 \leq P(E) \leq 1$;

Axiom 2

$P(\Omega) = 1$ (a certain event has probability 1);

Axiom 3

$P(E_1 \cup E_2) = P(E_1)+P(E_2)$, if $E_1 \cap E_2 = \emptyset.$

From the basic rules the following properties can be derived:

1:

$P(E) = 1 - P(\overline E)$;

2:

$P(\emptyset) = 0$;

3:

if $A\subseteq B$ then $P(A) \leq P(B)$;

4:

$P(A\cup B) = P(A) + P(B) - P(A\cap B)$.

We also anticipate here another rule which will be discussed in Section :

5:

$P(A\cap B) = P(A\,\vert\,B)\cdot P(B) = P(A)\cdot P(B\,\vert\,A)\,.$