Non-Naive Probability

Non-Naive Definition of Probability

Definition

• A probability space consists of a sample space $S$ and a probability function $P$ which takes an event $A\subseteq S$ as input and return $P(A)$ a real number between 0 and 1, as output.

Axioms for P

1. $P(\varnothing )=0,P(S)=1$
2. a

Properties of Probabilities

1. $P(A^c)=1-P(A)$
2. If $A\subseteq B$, then $P(A)\le P(B)$
3. $P(A\cup B)=P(A)+P(B)-P(A\cap B)$