Foundations of probability theory: basic definitions

Basic definitions

The basis of probability theory is a set of events - sample space - and a systematic set of numbers - probabilities -assigned to each event. The key aspect of the theory is the system of assigning probabilities. Formally, a sample space is the set $$ of all possible outcomes ${}_i$ of an experiment. An event is a collection of sample points ${}_i$ determined by some set-algebraic rules governed by the laws of Boolean algebra.Letting $A$ and $B$ denote events, these laws are $$\text{Union:}A\cup B=\{\colon \in A\lor \in B\}$$ $$\text{Intersection:}A\cap B=\{\colon \in A\land \in B\}$$ $$\text{Complement:}(A)=\{\colon \notin A\}$$ $$(A\cup B)=(A)\cap (B)$$ The null set $\emptyset$ is the complement of $$ . Events are said to be mutually exclusive if there is no element common to both events: $A\cap B=\emptyset$ .

Associated with each event $A_i$ is a probability measure $(A_i)$ , sometimes denoted by ${}_i$ , that obeys the axioms of probability .

• $(A_i)\ge 0$
• $()=1$
• If $A\cap B=\emptyset$ , then $(A\cup B)=(A)+(B)$ .

Suppose $(B)\neq 0$ . Suppose we know that the event $B$ has occurred; what is the probability that event $A$ has also occurred? This calculation is known as the conditional probability of $A$ given $B$ and is denoted by $(B, A)$ . To evaluate conditional probabilities, consider $B$ to be the sample space rather than $$ . To obtain a probability assignment under these circumstances consistentwith the axioms of probability, we must have