Probability Postulates

Let S denote the sample space of a random experiment, Oi, the basic outcomes, and A an event. For each event A of the sample space, S, we assume that P(A) is defined and we have the following probability postulates:

Consequences of the Postulates

We now list and illustrate some immediate consequences of the three postulates.

Example: A charitable organization sells 1,000 lottery tickets. There are 10 major prizes and 100 minor prizes, all of which must be won. The process of choosing winners is such that at the outset each ticket has an equal chance of winning a major prize, and each has an equal chance of winning of wining a minor prize. No ticket can win more than  one prize. What is the probability of winning a major prize with a single ticket? What is the probability of winning a minor prize? What is the probability of winning a minor prize? What is the probability of winning some prize?

Solution: Of the 1,000 tickets, 10 will win major prize, 100 will win minor prizes, and 890 will win no prize. Our single ticket is selected from the 1,000. Let A be the event “Selected ticket wins a major prize,” and let B be the event “Selected ticket wins a minor prize.” The probabilities are

        
The event “Ticket wins some prize” is the union of events A and B. Since only one prize is permitted, these events are mutually exclusive, and

                        

Example: Oil Well Drilling (Probability)
In the early stages of the development of the Hibernia oil site in the Atlantic Ocean, the Petroleum Directorate of Newfoundland estimated the probability to be 0.1 that economically recoverable reserves would exceed 2 billion barrels. The probability for reserves in excess of 1 billion barrels was estimated to be 0.5. Given this information, what is the estimated probability of reserves between 1 and 2 billion barrels?
Solution: Let A be the event “Reserves exceed 2 billion barrels” and B the event “Reserves between 1 and 2 billion barrels.” These are mutually exclusive, and their union, , is the event “Reserves exceed 1 billion barrels.” We therefore have