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Math 1342. Notes on Chapter 5
Some Definitions:
The sample space, S, of a probability experiment is the collection of all possible
outcomes.
Problem: We flip a coin twice. Write the sample space for the experiment.
S=
An event is any collection of outcomes from a probability experiment.
Example: When flipping a die, write the event of having an odd number.
A=
; The event of having a number less than 4:
; The event of having a number less than 7:
; The event of having a number more than 7:
D=
Rules of Probabilities: 0  P( Event )  1 ; and

P( E )  1 .
for all E ' s
Problem: Verify that the following is a probability model. What do we call the
outcome “blue”?
Color
Red
Green
Blue
Brown
Yellow
Orange
Probability
0.3
0.15
0
015
0.2
0.2
1
Problem: Let the sample space be S = 1,2,3,4,5,6,7,8,9,10 . Suppose the outcomes
are equally likely.
a. Compute the probability of the event E  1,2,3.
b. Compute the probability of the event F  3,5,9,10.
c. Compute the probability of the event G = “an even number less than 9.”
d. Compute the probability of the event H = “an odd number.”
Definition: Two events are disjoint or mutually exclusive if they have no
outcomes in common. Events E and F in the above example are disjoint.
If E and F are disjoint (or mutually exclusive) events, then
P(E or F) = P(E) + P(F)
If E and F are not disjoint, then P(E or F) = P(E) + P(F) – P(E and F).
Example: The following data represent the number of live multiple-delivery births
in 2005 for women 15 to 54 years old.
Age
15-19
20-24
25-29
30-34
35-39
40-44
45-54
# of Multiple
births
83
465
1635
2443
1604
344
120
Total
6694
2
(a) Determine the probability that a randomly selected multiple birth in 2005
for women 15 to 54 years old involved a mother 30 to 39 years old.
(b) Determine the probability that a randomly selected multiple birth in 2005
for women 15 to 54 years old involved a mother who was not 30 to 39
years old.
(c) Determine the probability that a randomly selected multiple birth in 2005
for women 15 to 54 years old involved a mother who was less than 45 years
old.
(d) Determine the probability that a randomly selected multiple birth in 2005
for women 15 to 54 years old involved a mother who was at least 20 years
old.
Problem: A guidance counselor at a middle school collected the following
information regarding the employment status of married couples within his
school’s boundaries.
Number of Children under 18 Years Old
Worked
Husband only
Wife only
Both spouses
Total
0
1
2 or More
Total
172
94
522
788
79
17
257
353
174
15
370
559
425
126
1149
1700
3
(a) What is the probability that, for married couple selected at random, both
spouses work?
(b) What is the probability that, for married couple selected at random, the
couple has one child under the age of 18?
(c) What is the probability that, for married couple selected at random, the
couple has two or more children under the age of 18 and both spouses work?
(d) What is the probability that, for married couple selected at random, the
couple has no children or only the husband works?
(e) Would it be unusual to select a married couple at random for which only the
wife works?
Multiplication Rule for Independent Events
If E and F are independent events, then P (E and F) = P (E) .P (F)
Problem: The probability that a randomly selected 40-year-old male will live to be
41 years old is 0.99757.
(a) What is the probability that two randomly selected 40-year-old males will
live to be 41 years old?
(b) What is the probability that five randomly selected 40-year-old males will
live to be 41 years old?
(c) What is the probability that at least one of five randomly selected 40-yearold males will not live to be 41 years old? Would it be unusual if at least one
of five randomly selected 40-years-old males did not live to be 41 years old?
Conditional Probability
The notation P(F/E) is read “the probability of event F given event E.” It is the
probability that the event F occurs, given that the event E has occurred.
If E and F are any two events, then
P( F / E ) 
P( EandF )
P( E )
Problem: For the month of June in the city of Chicago, 37% of the days are
cloudy. Also in the month of June in the city of Chicago, 21% of the days are
cloudy and rainy. What is the probability that a randomly selected day in June will
be rainy if it is cloudy?
4
Problem: The following data represent, in thousands, the type of health insurance
coverage of people by age in the year 2006.
Age<18
47,906
Private
health
insurance
Government 22,109
health
insurance
No health
8,661
insurance
Total
78,676
18-44
74,375
45-64
57,505
>64
21,904
Total
201,690
12,375
11,304
33,982
80,270
27,054
10,737
541
46,993
114,304
79,546
56,427
328,953
Determine P (<18 years old) and P (<18 years old/no health insurance). Are the
events “<18 years old” and “no health insurance” independent?
Example: Evaluate
9
p4
9
C2
12
C3
5