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Business Statistics:
A Decision-Making Approach
7th Edition
Chapter 4
Using Probability and
Probability Distributions
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 4-1
Important Terms




Random Variable - Represents a possible numerical value
from a random event and it can vary from trial to trial
Probability – the chance that an uncertain event will occur
(always between 0 and 1)
Experiment – a process that produces outcomes for
uncertain events
Sample Space (or event) – the collection of all possible
experimental outcomes
Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall
4-2
Basic Rule of Probability
Individual Values
0 ≤ P(Ei) ≤ 1
For any event Ei
The value of a probability is
between 0 and 1
0 = no chance of occurring
1 = 100% change of occurring
Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall
Sum of All Values
k
 P(e )  1
i 1
i
where:
k = Number of individual
outcomes in the
sample space
ei = ith individual outcome
The sum of the probabilities of all
the outcomes in a sample space
must = 1 or 100%
4-3
Simple probability


The probability of an event is the number of
favorable outcomes divided by the total number
of possible outcomes.
This assumes the outcomes are all equally
weighted.
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 4-4
Visualizing Events

Contingency Tables
Ace

Not Ace
Total
Black
2
24
26
Red
2
24
26
Total
4
48
52
Tree Diagrams
2
Sample
Space
Full Deck
of 52 Cards
Sample
Space
24
2
24
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 4-5
Simple probability

What is the probability that a card drawn at
random from a deck of cards will be an ace?




52 cards in the deck
4 are aces
The probability is 4/52
Each card represents a possible outcome  52
possible outcomes.
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 4-6
Simple probability
There are 36
possible outcomes
when a pair of dice
is thrown.
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 4-7
Simple probability

Calculate the probability that the sum of the two
dice will equal 5?


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Four of the outcomes have a total of 5: find out
from the table
(1,4; 2,3; 3,2; 4,1)
Probability of the two dice adding up to 5 is
4/36 = 1/9 since there are 36 possibilities.
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 4-8
Simple probability

Calculate the probability that the sum of the
two dice will equal 12?


Only one (6,6)
1/36
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 4-9
Notation of Probability


The probability of event A is denoted by P(A)
Example: Suppose a coin is flipped 3 times. What is
the probability of getting two tails and one head?





The sample space consists of 8 sample points.
S = {TTT, TTH, THT, THH, HTT, HTH, HHT, HHH}
the probability of getting any particular sample point
is 1/8.
getting two tails and one head: A = {TTH, THT, HTT}
P(A) = 1/8 + 1/8 + 1/8 = 3/8
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 4-10
Probability Concepts

Independent and Dependent Events


Independent: Occurrence of one does not
influence the probability of
occurrence of the other
Dependent: Occurrence of one affects the
probability of the other
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 4-11
Independent vs. Dependent
Events

Independent Events
A = heads on one flip of fair coin
B = heads on second flip of same coin
Result of second flip does not depend on the result of
the first flip.

Dependent Events
X = rain forecasted on the news
Y = take umbrella to work
Probability of the second event is affected by the
occurrence of the first event
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 4-12
Rule of Multiplication

The rule of multiplication applies to the situation when
we want to know the probability that two events (Event
A and Event B) both occur.

Independent: P(A ∩ B) = P(A) P(B)

Dependent: P(A ∩ B) = P(A) P(B|A)
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 4-13
Independent Events


If A and B are independent, then the probability
that events A and B both occur is: p(A and B)
= p(A) x p(B)
What is the probability that a coin will come up
with heads twice in a row?
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

Two events must occur: a head on the first toss
and a head on the second toss.
The probability of each event is 1/2
the probability of both events is: 1/2 x 1/2 = 1/4.
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 4-14
Independent Events

What is the probability that the first card is the
ace (put it back: replace) of clubs and the
second card is a club (any club)?



The probability of the first event is 1/52.
The probability of the second event 13/52 = 1/4
(composed of clubs)
1/52 x 1/4 = 1/208
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 4-15
Dependent Events


If A and B are not independent, then the
probability of A and B both occur is:
p(A and B) = p(A) x p(B|A)
where p(B|A) is the conditional probability of
B given A.
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 4-16
What is Conditional Probability?


A conditional probability is the probability of an
event given that another event has occurred.
Conditional probability for any
two events A , B:
P(A and B)
P(A | B) 
P(B)
where
P(B)  0
Notation
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 4-17
Conditional Probability Example
What is the probability that a car has a CD player, given that
it has AC ? we want to find P(CD | AC)
CD
No CD
Total
AC
.2
.5
.7
No AC
.2
.1
.3
Total
.4
.6
1.0
P(CD and AC) .2
P(CD | AC) 
  .2857
P(AC)
.7
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 4-18
Conditional Probability Example
(continued)

What is the probability that the total of two
dice will be greater than 8 given that the first
die is a 6?

This can be computed by considering only outcomes
for which the first die is a 6. Then, determine the
proportion of these outcomes that total more than 8.
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 4-19
Conditional Probability Example
(continued)
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There are 6 outcomes
for which the first die
is a 6.
And of these, there
are four that total
more than 8.
(6,3; 6,4; 6,5; 6,6)
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 4-20
Conditional Probability Example
(continued)

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6 outcomes for which the first die is a 6
There are four that total more than 8
The probability of a total greater than 8 given
that the first die is 6 is 4/6 = 2/3.
More formally, this probability can be written as:
p(total>8 | Die 1 = 6) = 2/3(6,3; 6,4; 6,5; 6,6).
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 4-21
Conditional Probability Example
(continued)

What is the probability that the first card is the
ace (not put it back: no replace) and the second
card is also an ace?
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First: p(A) = 4/52 = 1/13.
Of the 51 remaining cards, 3 are aces. So, p(B|A)
= 3/51 = 1/17
Probability of A and B is: 1/13 x 1/17 = 1/221.
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 4-22
Probability Concepts
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Mutually Exclusive Events
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If A occurs, then B cannot occur

A and B have no common elements
A
Black
Cards
B
Red
Cards
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
A card cannot be
Black and Red at
the same time.
Chap 4-23
Rule of Addition
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We have two events, and we want to know the
probability that either event occurs.
Mutually exclusive:

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P(A or B) = P(A) + P(B)
Not Mutually exclusive:

P(A or B) = P(A)+ P(B) – P(A and B)
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 4-24
Mutually Exclusive Events


If events A and B are mutually exclusive, then
the probability of A or B is p(A or B) = p(A) +
p(B)
What is the probability of rolling a die and
getting either a 1 or a 6?


impossible to get both a 1 and a 6 (that is, mutually
exclusive).
p(1 or 6) = p(1) + p(6) = 1/6 + 1/6 = 1/3
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 4-25
Mutually Exclusive Events
(continued)
A bag of candy contains 4 different
flavors: lemon, strawberry,
orange, and blueberry
What is the probability that you will
get a lemon OR an orange
piece of candy?
Flavor
Count
Lemon
8
Strawberry
12
Orange
5
Blueberry
10
TOTAL
35
E1 = lemon = (8/35) = 0.23
E2 = orange = (5/35) = 0.14
P(E1 or E2) = P(E1) + P(E2) = 0.23 + 0.14 = 0.37
Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall
4-26
Not Mutually Exclusive Events
■
If the events are not mutually exclusive,
P(A or B) = P(A)+ P(B) – P(A and B)
A
+
B
=
A
B
P(A or B) = P(A) + P(B) - P(A and B)
Don’t count common
elements twice!
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 4-27
Not Mutually Exclusive Example

What is the probability that a card will be either an ace
or a spade?
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p(ace) = 4/52 and p(spade) = 13/52
The only way both (ace and a spade) can be drawn is to draw
the ace of spades.
There is only one ace of spades: p(ace and spade) = 1/52 .
The probability of an ace or a spade can be:
p(ace)+p(spade)-p(ace and spade) =
4/52 + 13/52 - 1/52 = 16/52 = 4/13
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 4-28
Complement Rule

The complement of an event E is the collection of
all possible elementary events not contained in
event E. The complement of event E is
represented by E.
E

Complement Rule:
P(E)  1  P(E)
E
Or,
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
P(E)  P(E )  1
Chap 4-29
Probability Types
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Marginal probability
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Joint probability

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Involves two or more random variables, in which the outcome of
all is uncertain
Conditional probability

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Involves only a single random variable, the outcome of which is
uncertain
Involves two or more random variables, in which the outcome of
at least one is known
Example: download “Probability Type Example ” Excel
file
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 4-30
Chapter Summary

Discussed probability terminology

Described approaches to determining probabilities
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Reviewed common rules of probability

Addition Rules (Rules 1 – 5)

Multiplication Rules (Rules 8, 9)
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Defined conditional probability (Rules 6, 7)

Used Bayes’ Theorem for conditional probabilities
Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall
4-31