business statistics: a decision-making approach, 7th edition

Business Statistics

Department of Quantitative Methods & Information Systems

Dr. Mohammad Zainal QMIS 120

Chapter 4

Using Probability and

Probability Distributions

Chapter Goals

After completing this chapter, you should be able to:

Explain basic probability concepts and definitions

Use a Venn diagram or tree diagram to illustrate simple probabilities

Apply common rules of probability

Explain three approaches to assessing

probabilities

Apply common rules of probability, including the

Addition Rule and the Multiplication Rule

QMIS 120, by Dr. M. Zainal Chap 4-2

QMIS 120, by Dr. M. Zainal

Chapter Goals

Compute conditional probabilities

Determine whether events are statistically independent

Use Bayes’ Theorem for conditional probabilities

(continued)

Chap 4-3

Important Terms

Random Experiment – a process leading to an

uncertain outcome

Basic Outcome (Simple Event ) – a possible

outcome of a random experiment

Sample Space – the collection of all possible

outcomes of a random experiment

Event (Compound Event) – any subset of basic

outcomes from the sample space

Probability – the chance that an uncertain event

will occur (always between 0 and 1)



Sample Space

The Sample Space is the collection of all

possible outcomes

e.g., All 6 faces of a die:

e.g., All 52 cards of a bridge deck:

Chap 4-5


Sample Space

e.g., All 2 faces of a coin:

e.g., All 4 colors in the jar:

Chap 4-6

Examples of Sample Spaces

Rolling one die:

S = [1, 2, 3, 4, 5, 6]

Tossing one coin:

S = [Head, Tail]


Choosing one ball:

S = [Red, Blue, Green, Yellow]

Playing a game:

S = [Win, Lose, Tie]

Examples of Sample Spaces



Events

Experimental outcome – An outcome from a

sample space with one characteristic

Example: A red card from a deck of cards

Event – May involve two or more outcomes

simultaneously

Example: An ace that is also red from a deck of

cards

Chap 4-9


Visualizing Events

Contingency Tables

Tree Diagrams

Red 2 24 26

Black 2 24 26

Total 4 48 52

Ace Not Ace Total

Full Deck

of 52 Cards

Sample

Space

Sample

Space 2

24

2

24

Chap 4-10


Experimental Outcomes

A automobile consultant records fuel type and vehicle type for a sample of vehicles

2 Fuel types: Gasoline, Diesel

3 Vehicle types: Truck, Car, SUV

6 possible experimental outcomes:

e1 Gasoline, Truck

e2 Gasoline, Car

e3 Gasoline, SUV

e4 Diesel, Truck

e5 Diesel, Car

e6 Diesel, SUV

Chap 4-11

Important Terms

Intersection of Events – If A and B are two

events in a sample space S, then the

intersection, A ∩ B, is the set of all outcomes in

S that belong to both A and B

A B AB

S


Important Terms

A and B are Mutually Exclusive Events if they

have no basic outcomes in common

i.e., the set A ∩ B is empty

If A occurs, then B cannot occur

A and B have no common elements

(continued)

A B

S


A ball cannot be

Green and Red

at the same

time.

Important Terms

Union of Events – If A and B are two events in a

sample space S, then the union, A U B, is the

set of all outcomes in S that belong to either

A or B

(continued)

A B

The entire shaded area

represents A U B

S


Important Terms

Events E1, E2, … Ek are Collectively Exhaustive

events if E1 U E2 U . . . U Ek = S

i.e., the events completely cover the sample space

The Complement of an event A is the set of all

basic outcomes in the sample space that do not

belong to A. The complement is denoted

= +

(continued)

A

A S

A AA

A


Examples

Rolling a die

S = [1, 2, 3, 4, 5, 6]

Let A be the event “Number rolled is even”

Let B be the event “Number rolled is at least 4”

Then:


(continued)

Complements:

Intersections:

Unions:

Examples


Mutually exclusive:

Collectively exhaustive:

(continued)

Examples


Probability

Probability – the chance that

an uncertain event will occur

(always between 0 and 1)

Also, it is a numerical

measure of the likelihood

that an event will occur

0 ≤ P(A) ≤ 1 For any event A

Certain

Impossible

.5

1

0


Assessing Probability

There are three approaches to assessing the

probability of an uncertain event:

1. classical probability

Assumes all outcomes in the sample space are

equally likely to occur


spacesampletheinoutcomesofnumbertotal

eventthesatisfythatoutcomesofnumber

N

NAeventofyprobabilit A

S = [1, 2, 3, 4, 5, 6] A = [2, 4, 6] C = [5, 6]

Classical Probability Examples


Example: Find the probability of obtaining an

even number in one roll of a die and find the

probability of obtaining a number greater than

4.

Counting the Possible Outcomes

Use the Combinations formula to determine the

number of distinct combinations of n distinct objects

that can be formed, taking them r at a time (No

ordering)

where

n! = n(n-1)(n-2)…(1)

0! = 1 by definition

!( )

!( )!

n n

n r r r

nC C

r n r



Combinations: If one has 5 different objects (e.g. A, B,

C, D, and E), how many ways can they be grouped as

3 objects when position does not matter (e.g. ABC,

ABD, ABE, ACD, ACE, ADE are correct but CBA is

not ok as is equal to ABC)



Use the Permutations formula to determine the

number of ways we can arrange n distinct objects,

taking them r at a time (with ordering)

where

n! = n(n-1)(n-2)…(1)

0! = 1 by definition

!( )

( )!

n n

n r r r

nP P

n r



Permutations: Given that position (order) is important,

if one has 5 different objects (e.g. A, B, C, D, and E),

how many unique ways can they be placed in 3

positions (e.g. ADE, AED, DEA, DAE, EAD, EDA,

ABC, ACB, BCA, BAC etc.)


Assessing Probability

Three approaches (continued)

2. relative frequency probability

the limit of the proportion of times that an event A occurs in a large

number of trials, n

3. subjective probability

an individual opinion or belief about the probability of occurrence

populationtheineventsofnumbertotal

Aeventsatisfythatpopulationtheineventsofnumber

n

nAeventofyprobabilit A


Relative Frequency Probability

Examples


Example: In a group of 500 women, 80 have played

football at least once in their life. Suppose one of these

500 woman is selected. What is the probability that she

played football at least once?

Solution: n = 500 and f = 80

Relative frequency f Women

420 / 500 = 0.84 420 Did not play football

80 / 500 = 0.16 80 Played football

Sum = 1.0 n = 500

Probability Postulates

1. If A is any event in the sample space S, then

2. Let A be an event in S, and let Ei denote the basic

outcomes. Then

(the notation means that the summation is over all the basic outcomes in A)

3. P(S) = 1

1P(A)0

i

A

P(A) P(E )


Probability Rules

The Complement rule:

1)AP(P(A)i.e., P(A)1)AP(

A

A


Joint and Marginal Probabilities

The probability of a joint event, A ∩ B:

Computing a marginal probability:

Where B1, B2, …, Bk are k mutually exclusive and

collectively exhaustive events

outcomeselementaryofnumbertotal

BandAsatisfyingoutcomesofnumberB)P(A

)BP(A)BP(A)BP(AP(A) k21



Marginal Probability

Marginal probability is the probability of a single

event without consideration of any other event

Suppose one employee is selected out of a

sample of 100, he/she maybe classified either on

the bases of gender alone or on the bases of a

marital status.

Total Single Married

60 45 15

40 36 4

Male

Female

Total 100 81 19

Chap 4-31


Marginal Probability Example

The probability of each of the following event is

called marginal probability


60 45 15

40 36 4

Male

Female

Total 100 81 19

P(male) = 60/100

P(female) = 40/100

P(married) = 19/100

P(single) = 81/100

Chap 4-32


Conditional Probability

This probability is called conditional probability and

is written as “the probability that the employee

selected is married given that he is a male.”

P(married | male )

Read as “given”

This event has

already occurred

The event whose probability

is to be determined

Chap 4-33

Conditional Probability

A conditional probability is the probability of one

event, given that another event has occurred:

P(B)

B)P(AB)|P(A

P(A)

B)P(AA)|P(B

The conditional

probability of A

given that B has

occurred

The conditional

probability of B

given that A has

occurred



Example: Find the conditional probability P(married | male)

for the data on 100 employees

Solution


60 45 15 Male

40 36 4 Female

100 81 19 Total

Conditional Probability Example

Chap 4-35


(continued)


Chap 4-36


What is the probability that a car has a CD player, given that it has AC ?

i.e., we want to find P(CD | AC)

Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both.


Chap 4-37


No CD CD Total

AC

No AC

Total 1.0

Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD).

20% of the cars have both.

(continued)


Chap 4-38

Using a Tree Diagram

All

Cars

Given AC

or no AC:




Example: Consider the experiment of tossing a fair die.

Denote by A and B the following events:

A={Observing an even number}, B={Observing a number of

dots less than or equal to 3}. Find the probability of the

event A, given the event B.

Solution

Chap 4-40

Multiplication Rule

Multiplication rule for two events A and B:

also

P(B)B)|P(AB)P(A

P(A)A)|P(BB)P(A



Independent and Dependent Events

Independent: Occurrence of one does not

influence the probability of

occurrence of the other

Probability Concepts

E1 = heads on one flip of fair coin

E2 = heads on second flip of same coin

Result of second flip does not depend on the

result of the first flip.

Chap 4-42


Independent and Dependent Events

Dependent: Occurrence of one affects the

probability of the other

Probability Concepts

E1 = rain forecasted on the news

E2 = take umbrella to work

Probability of the second event is affected by the

occurrence of the first event

Chap 4-43

Statistical Independence

Two events are statistically independent

if and only if:

Events A and B are independent when the probability of one

event is not affected by the other event

If A and B are independent, then

P(A)B)|P(A

P(B)P(A)B)P(A

P(B)A)|P(B

if P(B)>0

if P(A)>0



Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both.

Statistical Independence Example

No CD CD Total

AC .2 .5 .7

No AC .2 .1 .3

Total .4 .6 1.0

Are the events AC and CD statistically independent?

Chap 4-45

No CD CD Total

AC .2 .5 .7

No AC .2 .1 .3

Total .4 .6 1.0

(continued)

Statistical Independence Example



For Independent Events:

Conditional probability for

independent events E1 , E2:

)P(E)E|P(E 121 0)P(Ewhere 2

)P(E)E|P(E 212 0)P(Ewhere 1

Chap 4-47

Probability Rules

The Addition rule:

The probability of the union of two events is

B)P(AP(B)P(A)B)P(A

A B

P(A or B) = P(A) + P(B) - P(A and B)

Don’t count common

elements twice!

A B + =



Addition Rule for Mutually Exclusive Events

If A and B are mutually exclusive, then

P(A ∩ B) = 0

So

P(A or B) = P(A) + P(B) - P(A and B)

P(A U B) = P(A) + P(B)

A B

Chap 4-49

A Probability Table

B

A

A

B

)BP(A

)BAP( B)AP(

P(A)B)P(A

)AP(

)BP(P(B) 1.0P(S)

Probabilities and joint probabilities for two

events A and B are summarized in this table:


Law of Total Probability

Consider the following Venn diagram

Can we find the area (probability) of A assuming

that we know the probability of each Bi & P(A|Bi) ?

B1 B2 Bn-1 Bn

B

A

A1 A2 An-1 An


Bayes’ Theorem

where:

Bi = ith event of k mutually exclusive and collectively

exhaustive events

A = new event that might impact P(Bi)

i i

1 1 2 2

P(A|B )P(B )

P(A|B )P(B ) P(A|B )P(B )

i

i

P(A B )P(B |A)

P(A)



Bayes’ Theorem Example

A drilling company has estimated a 40%

chance of striking oil for their new well.

A detailed test has been scheduled for more

information. Historically, 60% of successful

wells have had detailed tests, and 20% of

unsuccessful wells have had detailed tests.

Given that this well has been scheduled for a

detailed test, what is the probability that the well will be successful?

Chap 4-53

Let S = successful well

U = unsuccessful well

P(S) = , P(U) = (prior probabilities)

Define the detailed test event as D

Conditional probabilities:

Goal is to find

(continued)



Apply Bayes’ Theorem:

(continued)




Given the detailed test, the revised probability

of a successful well has from the original

estimate of

Bayes’ Theorem Example (continued)

Chap 4-56

Complete Example

Consider a standard deck of card that we see normally

being played on computer or table. It consists of 52

cards with 4 suits and so 13 cards in a suit.

Consider a standard deck of 52 cards, with four suits:

Spade (S):♠ Heart (H): ♥

Club (C): ♣ Diamond (D): ♦

Spades and Clubs are traditionally

in black color, while Hearts and

Diamonds are in red.

Let event A = card is an Ace

Let event B = card is from a Red suit



Black

Color Type Red Total

Ace 2 2 4

Non-Ace 24 24 48

Total 26 26 52

What is the probability of Red given it was Ace (Cond.)

Chap 4-58

Complete Example

Black


Ace 2 2 4

Non-Ace 24 24 48

Total 26 26 52

What is the probability of Red and Ace (Intersection)


Complete Example

Black


Ace 2 2 4

Non-Ace 24 24 48

Total 26 26 52

What is the probability of Red or Ace (Union)


Complete Example

Black


Ace 2 2 4

Non-Ace 24 24 48

Total 26 26 52

What is the probability of Ace (Marginal)


Complete Example

Black


Ace 2 2 4

Non-Ace 24 24 48

Total 26 26 52

Are the events Red and Ace statistically independent?


Complete Example

Black


Ace 2 2 4

Non-Ace 24 24 48

Total 26 26 52

What is the complement of the event Ace?


Complete Example

Example: Manufacturing firm that receives shipment of parts from two different suppliers. Currently, 65 percent of the parts purchased by the company are from supplier 1 and the remaining 35 percent are from supplier 2. Historical Data suggest the quality rating of the two supplier are shown in the table:

Good Parts Bad Parts

Supplier 1 98 2

Supplier 2 95 5

a) Draw a tree diagram for this experiment with the probability

of all outcomes

b) Given the information the part is bad, What is the

probability the part came from supplier 1?



Apply Bayes’ Theorem:

(continued)



Example: An insurance company rents 35% of the cars for

its customers from Avis and the rest from Hertz. From past

records they know that 8% of Avis cars break down and 5%

of Hertz cars break down.

A customer calls and complains that his rental car broke

down. What is the probability that his car was rented from

Avis?



Chapter Summary

Defined basic probability concepts

Sample spaces and events, intersection and union

of events, mutually exclusive and collectively

exhaustive events, complements

Examined basic probability rules

Complement rule, addition rule, multiplication rule

Defined conditional, joint, and marginal probabilities

Reviewed odds and the overinvolvement ratio

Defined statistical independence

Discussed Bayes’ theorem


Copyright

The materials of this presentation were mostly

taken from the PowerPoint files accompanied

Business Statistics: A Decision-Making Approach,

7e © 2008 Prentice-Hall, Inc.


business statistics: a decision-making approach, 7th edition

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