Properties of Probabilities 9/28

We divide the properties into two groups:

Basic Axioms of Probability:
(a) $0\le P(A)\le 1$ for any set A.
(b) $P(\Omega)=1$, where $\Omega$ is the sample space.
(c) If $E_1, E_2,\ldots E_n$ is a sequence of mutually exclusive events, that is for all i and j different: $E_i E_j= \phi$ we have the finite additivity property:

$$P(\cup_{i=1}^{n}E_i)=\sum_{i=1}^{n}P(E_i)$$

Consequences:
Property 1:

$$P(\Omega)=P(E\cup E^c)=P(E)+P(E^c)$$

P(E^c)=1-P(E)

Property 2:

$$\mbox{If } E \subset F, P(E)\leq P(F).$$

Property 3:

$$P(E\cup F)=P(E)+P(F)-P(E \cap F).$$

Special Case: If E and F are disjoint, (mutally exclusive), then:

$$P(E\cup F)=P(E)+P(F)$$

Property 4:
If $F \subset E$

$$P(E\backslash F)=P(E\cap F^c)=P(E)-P(F)$$

Definition:
We define a  partition of the sample space $\Omega$ to be a sequence of pairwise disjoint sets $A_1,A_2,\dots,A_n$ whose union is $\Omega$.   Property 5:
If $A_1,A_2,\dots,A_n$ forms a partition of $\Omega$ then:

$$P(E)=\sum_{i=1}^n P(E\cap A_i)$$

Property 6:

$$P(A)=P(A\cap B)+P(A\cap B^c)$$

Property 7:

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