1. Axioms, Interpretations and Properties
Probability
Axioms, Interpretations and Properties

2. Objective of probability
Given an experiment and a sample space S, the objective of probability is to assign to each event A a number P(A) that gives a precise measure of the chance that A will occur.
To ensure that these assignments will be consistent with intuitive notions of probability, all assignments should satisfy the following basic axioms .

3. Axioms of probability For any event A, . P(S)=1
If is an infinite collection of disjoint events,

4. Properties of probability
Proof: Consider , disjoint sets with Then by axiom 3,
, which implies that
Also if are disjoint, append
Then

5. Another property of probability
For an event , , and thus .
Proof: Since and the sets are disjoint,

6. A property of probability
For any event ,
Proof:
by Axiom 1.

7. Probability for a union of events
For any events and ,
Proof: Since (which are disjoint sets),

8. Application
In a certain residential suburb, 60% of all households get Internet service from the local cable company, 80% get television from that company, and 50% get both. What is the probability that a randomly selected household gets at least one of the two services from the company, and what is the probability that they get exactly one?
The formula for the probability of the union of events is relevant, and the solutions become easy if we use a Venn diagram.

9. Probability for the union of three events
For any events A, B, and C,

10. Adding probabilities for simple events
Let denote the simple events of a sample space each consisting of a single outcome. Then the probability of a compound event A consisting of the simple events is:

11. Equally likely events
In some experiments consisting of N outcomes, it is reasonable to assign equal probability to all N simple events. Then
Thus

12. Objective probabilities
In an experiment that can be repeated, let n(A) be the number of occurrences of event A and n the number of repetitions of the experiment. Then n(A)/n is called the relative frequency of the event A.
As n gets large, this relative frequency converges to a limiting value that we identify with P(A).

13. Subjective probabilities
In some instances we make probabilistic statements about situations that are not repeatable. As an example: “It is likely that our company will be awarded the contract.”
Any assignment of probability in this case is subjective because different observers would have different prior information and different opinions on the likelihood.