1. Probability Measures: Axioms and Properties

2. Axioms for a Probability Measure
A probability measure assigns to each event E of a sample space S a number denoted by Pr[E], or P(E), and called the probability of E.
Pr[E] must be between 0 and 1 for each event
Pr[S] = 1
If E and F are disjoint events in S, then Pr[E U F] = Pr[E] + Pr[F]

3. Find Pr[high value sale] Find Pr[sale after 6:00 pm]
Outcome
Abbreviation
Probability
Before 6:00 and high value BH
.05
Before 6:00 and not high value BN
.26
After 6:00 and high value AH
.17
After 6:00 and not high value AN
.52
Find Pr[high value sale]
Find Pr[sale after 6:00 pm]
Find Pr[sale after 6:00 pm or high value]

4. Properties of a Probability Measure
For any event E, Pr[E] = 1 – Pr[E’]
For any events E and F,
Pr[E U F] = Pr[E] + Pr[F] – Pr[E ∩ F]
This follows from the sizes of sets formula we used in Chapter 1

5. Example
Let E and F be events in sample space S with Pr[E] = .65, Pr[F] = .4, and Pr[E ∩ F] = .3
Find Pr[E U F]
Find the probability of event G, where G is the set of all outcomes which are in exactly one of events E or F.
Look at Example 4 and 5 on p. 68

6. Classwork/Homework
Work on p. 70 #1 – 3, 13 – 15, 19 – 23, 25 – 27