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Chapter 5
A Survey of
Probability Concepts
Our Objectives
Define probability.
 Describe the classical, empirical, and
subjective approaches to probability.
 Understand the terms experiment,
event, and outcome.
 Define the terms conditional
probability and joint probability.

Our Objectives (cont’d)
Calculate probabilities using the rules
of addition and rules of multiplication.
 Use a tree diagram to organize and
compute probabilities.

Probability
A value between zero and one,
inclusive, describing the relative
possibility (chance or likelihood) an
event will occur.
 The probability of 1 represents
something that is certain to happen;
the probability of 0 represents
something that cannot happen.

Experiment in Statistics

A process that leads to the occurrence
of one and only one of several
possible observations.
Outcome in Statistics

A particular result of an experiment.
Event in Statistics

A collection of one or more outcomes
of an experiment.
Examples
Experiment: roll a die.
 An outcome: a 6.
 Event: observe a number greater than
3.

Approaches to Assigning
Probabilities

Objective Probability
Classical Probability
 Empirical Probability


Subjective Probability
Classical Probability
Based on the assumption that the
outcomes of an experiment are
equally likely.
 Probability of an event= Number of
favorable outcomes / Total number of
possible outcomes.

Example
We roll a die. What is the probability
of the event “an even # appears face
up”?
 Possible outcomes are:1,2,3,4,5,6.(6)
 Favorable outcomes are:2,4,6.(3)
 Probability of an even number=3/6
=.5

Sum of Classical Probabilities
If a set of events is mutually exclusive
and collectively exhaustive, then the
sum of the probabilities is 1.
 Mutually exclusive: occurrence of one
event means that none of the other
events can occur at the same time.
 Collectively exhaustive: at least one of
the events must occur when an
experiment is conducted.

Empirical Probability

Based on relative frequency.

Probability of an event= Number of
times event occurred in the past /
Total number of observations
Example
What is the probability of a future
space shuttle mission being
successful, given that 2 out of the last
113 missions ended with a disaster?
 Probability of a successful mission=
Number of successful flights / Total
number of flights.
 P(A)= 111 / 113= .98

Subjective Probability
The likelihood of a particular event
happening that is assigned by an
individual based on whatever
information is available.
 There is little or no past experience or
information on which to base a
probability.

Rules for Computing Probabilities

Rules of Addition

Special rule of addition:
P(A or B)=P(A) + P(B)
(the events must be mutually
exclusive)
Example
Weight
Event
Underweight
A
100
.025
Satisfactory
B
3,600
.900
Overweight
C
300
.075
4,000
1.000
Total
# of Packages
Probability of
Occurrence
What is the probability that a particular package will be
either underweight or overweight?
Example (cont’d)
The outcome underweight is event A.
 The outcome overweight is event C.
 P(A or C)= P(A) + P(C)
= .025 + .075
= .10

Venn Diagram

Graphically portrays the outcome of
an experiment.
Event
A
Event
B
Event
C
Complement Rule
P(A) + P(~A)= 1
Or
P(A)= 1 – P(~A): complement rule
Event
A
~A
Example
Use complement rule to show
probability of a satisfactory bag.
 Probability of unsatisfactory bag is P(A
or C)= P(A) + P(C)= .100
 Probability of satisfactory bag is
P(B)= 1 – [P(A) + P(C)]= 1 – .100= .900

C
A
.025
~(A or C)
.900
.075
General Rule of Addition
Used when outcomes of experiment
are not mutually exclusive.
 P(A or B)= P(A) + P(B) – P(A and B)

A
B
A and B
Joint Probability of A and B
Example
What is the probability of a card
chosen at random from a deck of
cards will be either a king or a heart?
 King: P(A)= 4/52
 Heart: P(B) = 13/52
 King of Hearts: P(A and B)= 1/52
 P(A or B)= P(A) + P(B) – P(A and B)

= 4/52 + 13/52 – 1/52
= .3077
Rules of Multiplication
Used to calculate probability of two
events happening.
 Special rule :

Used when 2 events are independent
(occurrence of one has no effect on
probability of occurrence of second).
 P(A and B)= P(A)P(B)

Rules of Multiplication (cont’d)




A survey of AAA revealed 60% of members
made airline reservations last year. Two
members are selected at random. What is
the probability both made reservations last
year?
1st member making reservation: P(R1)=.60
2nd member making reservation: P(R2)=.60
P(R1 and R2)=P(R1)P(R2)=(.60)(.60)=.36
Rules of Multiplication (cont’d)

If 2 events are dependent we use the
general rule of multiplication.

Conditional probability: the probability
of a particular event occurring, given
that another event has occurred.
General Rule of Multiplication
For 2 events, A and B, the joint
probability that both will occur is found
by multiplying the probability event A
will occur, by the conditional
probability of event B occurring, given
that A has occurred.
 P(A and B)= P(A)P(B|A)

Example




We have a box with 10 rolls of film, of which 3
are defective. What is the probability of getting a
defective roll the first time we draw, followed by
a defective roll the second time we draw
(assuming there are no replacements)?
1st film being defective: P(D1)=3/10 (3 out of 10
are defective)
2nd film being defective: P(D2|D1)= 2/9 (now 2
out of 9 are defective)
P(D1 and D2)=
P(D1)P(D2|D1)=(3/10)(2/9)=6/90=.07
Contingency Tables
A table used to classify sample
observations according to 2 or more
identifiable characteristics.
 Is a cross tabulation that
simultaneously summarizes 2 or more
variables of interest and their
relationship.

Example of Contingency Table
ATM
Male
withdrawals
per week
0
20
Female
Total
40
60
1
40
30
70
2 or more
10
10
20
Total
70
80
150
Example-Contingency Table Application
Loyalty
Remain, A
Not Remain, ~A
Total
B1
10
25
35
B2
30
15
45
B3
5
10
15
B4
75
30
105
Total
120
80
200
A sample of 200 executives were surveyed about their
loyalty to the company. They also indicated their years
of service with the company, B1 being <1year, B2 being
1-5 years, B3 6-10 years, and B4 >10 years.
What is the probability of randomly selecting an
executive who would remain or has less than 1 year of
service?
Example (cont’d)






Event A1: an executive selected at random
will remain with the company.
P(A1)= 120/200=.60 (120 out of 200 would
remain)
Event B1: an executive selected at random
has <1 year of service.
P(B1)= 35/200=.175
P(A1 and B1)=10/200=.05
P(A1 or B1)= P(A1)+P(B1)-P(A1 and B1)
= .60 + .175 - .05 = .725
Tree Diagram
A graph that is helpful in organizing
calculations involving several stages.
 The branches of a tree diagram are
weighted by probabilities.

Homework

12th edition:


48,49,50 (pg.171), 52,54,55 (pg.172),
67 (pg.174).
13th edition:

48, 49, 50, 52 (pg.172), 54,55
(pg.173), 66 (pg.174).