4.1 THE BASICS OF
PROBABILITY
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(Page 171)
Probability – the chance that a particular
event will occur.
The probability value will be from 0 to 1.
A value of 0 means that the event will not occur.
A probability of 1 means that the event will occur.
Anything between 0 and 1 reflects the uncertainty
of the event occurring.
CHANCE
Chance is how likely it is that something
will happen. To state a chance, we use a
percent.
0
Certain not
to happen
0%
½
Probability
Equally likely to
happen or not to
happen
Chance
50%
1
Certain to
happen
100%
Example:

Walking out in the road and you see
the rain coming:
The probability of the rain today is 1
since it is for sure that it will rain.
Example:
If an airplane has a top speed of 450 mph,
and the distance between city A and city
B is 900 miles:
The probability that the plane will make
the trip in 1.5 hours is 0.
city A
City B
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Experiment – a process that produces a
single outcome whose result cannot
be predicted with certainty.
Sample Space – is the collection of all
outcomes that can result from a
selection, decision, or experiment.
Flipping of 2 coins
Outcomes: Head and Tail
Sample Space:
1st flip – Head, 2nd flip – Head
1st flip – Head, 2nd flip – Tail
1st flip – Tail, 2nd flip – Head
1st flip – Tail, 2nd flip - Tail
(H, H)
(H, T)
(T, H)
(T, T)
The collection of possible experimental outcomes
is called the sample space.
Defining the Sample Space
Example 4.1 Page 171
Best –Bath Systems is interested in analyzing the
sales of its three main product lines.
Walk-in shower
Jacuzzi
Standard tub
Step 1. Define the Experiment
The experiment is the sale. The variable of interest
is the product sold.
Step 2. Defines the outcomes for one trial of the experiment.
e1 = walk-in shower
e2 = jacuzzi
e3 = standard tub
Step 3. Define the sample space.
For single sale: SS = (e1, e2, e3)
For two sales: SS = (e1, e2, e3, e4, e5, e6, e7, e8, e9)
Outcome
e1
e2
e3
e4
e5
e6
e7
e8
e9
Sale 1
walk-in
walk-in
walk-in
jacuzzi
jacuzzi
jacuzzi
standard
standard
standard
Sale 2
walk-in
jacuzzi
standard
walk-in
jacuzzi
standard
walk-in
jacuzzi
standard
Using a Tree Diagram to Define the
Sample Space
Example 4-2, Page 172
Lincoln Marketing Research was retained to interview
three television viewers to determine whether they
objected to having ads for hard liquor on TV. The
analyst assigned to the project is interested in
listing the sample space.
Step 1. Define the experiment.
The experiment involves selecting three television
viewers and posting the question: Would you object
to hard liquor advertisements on television?
Step 2. Determine the outcome.
Possible outcomes are:
Yes and No
Step 3. Define the sample Space.
Outcome
e1
e2
e3
e4
e5
e6
e7
e8
Viewer 1
no
no
no
no
yes
yes
yes
yes
Viewer 2
no
no
yes
yes
no
no
yes
yes
Viewer 3
no
yes
no
yes
no
yes
no
yes
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Tree Diagram
A tree diagram is often a useful way to define
the sample space for an experiment that helps
ensure that no outcomes are omitted.
Example 4 – 3
Using a Tree Diagram to Define the Sample
Space
Step 1. Define the experiment.
Step 2. Define the outcomes for a single
trial of the experiment.
Step 3. Define the sample space for three
trials using a tree diagram.
Viewer 1
Viewer 2
Viewer 3
No
No
No
Yes
Yes
No
Yes
No
Yes
No
Yes
Yes
No
Yes
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Exercises
4 -3 If two customers are asked to list their choice of
ice cream flavor from among vanilla, chocolate, and
strawberry, list the sample space showing the
possible outcomes.
Outcome
Customer 1 customer 2
e1
V
V
e2
V
C
e3
V
S
e4
C
V
e5
C
C
e6
C
S
e7
S
V
e8
S
C
e9
S
S
Customer 1
Customer 2
V
C
Vanilla
S
V
Chocolate
Strawberry
C
S
V
C
S
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Homework
Answer 4 - 4
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Event of Interest
Event – is a collection of experimental outcomes.
Example 4-3
Defining an Event of Interest
KPMG Accounting is interested in its audit status.
Step 1. Define the Experiment.
The experiment consists of two randomly
chosen audits.
Step 2. List the outcomes associated with one
trial of the experiment.
audit done early
audit done on time
Audit done late
Step 3. Define the sample space.
Outcome
e1
e2
e3
e4
e5
e6
e7
e8
e9
Audit 1
early
early
early
on time
on time
on time
late
late
late
Audit 2
early
on time
late
early
on time
late
early
on time
late
Step 4. Define the event of interest.
The event of interest at least one audit is
completed late is composed of all outcomes in
which one or more audits are late. This event
(E):
E = (e3, e6, e7, e8, e9)
There are five ways in which one or more audits are completed
late.
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Mutually Exclusive Events :
Two events are mutually exclusive if the
occurrence of one event precludes the occurrence
of the other event.
Example 4 – 4
Mutually Exclusive Events
Tech-Works, Inc. does contract assembly work for
Hewlett-Packard. Each item produced on the assembly
line can be thought of as an experimental trial. The
manager at this facility can analyze their process to
determine whether the events of interest are mutually
exclusive.
Step 1. Define the experiment.
The experiment is producing a
part on an assembly line.
Step 2. Define the outcomes for a single trial of
the experiment.
On each trial the outcome is
either a good or a defective item.
Step 3. Define the sample space.
Product 1
Product 2
e1
good
good
e2
good
defective
e3
defective
good
e4
defective
defective
Step 4. Determine whether the events are
mutually exclusive.
Let E1 = both products are good
Let E2 = at least one defective
E1 and E2 are mutually exclusive because
the two events have no outcomes in
common.
Having two good items and at the same time
having at least one defective item is not
possible.
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1.
Methods of Assigning Probability
Classical Probability – the method of
determining probability based on the ratio of
the number of ways an outcome or event of
interest can occur to the number of ways
any outcome or event can occur when the
individual outcomes are equally likely.
P(Ei) =
No. of ways Ei can occur
Total number of possible outcomes
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Example 4 – 5
Classical Probability Assessment
Galaxy Furniture: Each customer making a purchase exceeding
$100 will qualify to select an envelope that has coupons for
percentage discounts inside. There were 500 coupons. 400 were
for 10% discount, 50 were for 20% discount, 45 were for 30%
discount, and 5 were for 50% discount. Customers were
interested in determining the probability of getting a particular
discount amount.
Step 1. Define the experiment. (Envelope is selected from a large
drum)
Step 2. Determine whether the possible outcomes are equally likely.
Equally likely
Step 3. Determine the total number of outcomes. (500 envelopes)
Step 4. Define the event of interest. (First customer will get a 20%
discount)
Step 5. Determine the number of outcomes associated with the
event of interest. (50 coupons with a discount of 20%)
Step 6. Compute the classical probability.
Number of ways Ei can occur
P(Ei) = Total number of possible outcomes
P(20% discount) =
Number of ways 20% can occur
Total number of possible outcomes
=
50
500
= 0.10
After the first customer selects an envelope from
the drum, then probability that the next
customer will get a particular discount will
change, because the values in the
denominator and numerator will change.
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Many games of chance are based on classical
probability assessment.
Classical probability is difficult to apply to most
business situations.
Example: Starting a business:
SS = (Succeed, Fail)
Find the probability that the business will succeed:
P(Succeed) = ½
Is this true?
Many factors go into determining the success or
failure of a business. The possible outcome is not
equally likely.
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2. Relative Frequency Assessment– the method that
defines probability as the number of times an event
occurs divided by the total number of times an
experiment is performed in a large number of trials.
P(Ei) =
where:
Number of times Ei occurs
N
Ei = The event of interest
N = Number of trials
Relative Frequency Assessment approach is based
on actual observations.
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Example 4 – 6
Relative Frequency Probability Assessment
Starbucks coffee sells caffeinated and decaffeinated drinks.
The manager is interested in determining the probability
that a customer will select a caffeinated versus a decaf
drink.
Step 1. Define the experiment.
A randomly chosen customer will select
between decaf and caffeinated drink.
Step 2. Define the event of interest.
the manager is interested in the event E1
customer selects caffeinated.
Step 3. Determine the number of occurrences.
the manager has observed 2,250 sales of decaf and
caffeinated drinks. N = 2,250. There were 1,570 sales
for caffeinated drinks
Step 4. For the event of interest, determine the
number of occurrences.
There were 1,570 sales for caffeinated drinks.
P(Ei) =
No. of times Ei occur = 1,570
N = 2,250
= 0.6978
Based on the past history, the chance that a customer
will purchase a caffeinated drink is just under 70% or
a probability of 0.70.
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3. Subjective Probability – the method that defines
probability of an event as reflecting a decision
maker’s state of mind regarding the chances that
the particular event will occur.
A subjective probability is a measure of a personal
conviction that an outcome will occur.
The probability represents a person’s belief that an
event will occur.
Exercises: Page 180
1.
2.
3.
4.
Answer 4 – 5 Page 180
Answer 4 – 7 Page 180
Answer 4 – 8 Page 180
Answer 4 – 9 Page 180