Probability Measures:
Axioms and Properties
Axioms for a Probability
Measure
O 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.
O Pr[E] must be between 0 and 1 for each
event
O Pr[S] = 1
O If E and F are disjoint events in S, then
Pr[E U F] = Pr[E] + Pr[F]
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]
Properties of a Probability
Measure
O For any event E, Pr[E] = 1 – Pr[E’]
O For any events E and F,
Pr[E U F] = Pr[E] + Pr[F] – Pr[E ∩ F]
O This follows from the sizes of sets formula we
used in Chapter 1
Example
O Let E and F be events in sample space S
with Pr[E] = .65, Pr[F] = .4, and Pr[E ∩ F] = .3
O Find Pr[E U F]
O Find the probability of event G, where G is the
set of all outcomes which are in exactly one of
events E or F.
O Look at Example 4 and 5 on p. 68
Classwork/Homework
O Work on p. 70 #1 – 3, 13 – 15, 19 – 23,
25 – 27