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Introduction to ProbabilityUncertainty, Probability, Tree Diagrams,

Combinations and Permutations

Chapter 4BA 201

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PROBABILITY

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Uncertainty

Managers often base their decisions on an analysis of uncertainties such as the following:

What are the chances that sales will decreaseif we increase prices?

What is the likelihood a new assembly method method will increase productivity?

What are the odds that a new investment willbe profitable?

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Probability

Probability is a numerical measure of the likelihood that an event will occur.

Probability values are from 0 to 1.

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Probability as a Numerical Measureof the Likelihood of Occurrence

0 10.5

Increasing Likelihood of Occurrence

Probability:

The eventis veryunlikelyto occur.

The occurrenceof the event is

just as likely asit is unlikely.

The eventis almostcertain

to occur.

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STATISTICAL EXPERIMENTS

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Statistical Experiments

In statistical experiments, probability determines outcomes.

Even though the experiment is repeated in exactly the same way, an entirely different outcome may occur.

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An Experiment and Its Sample Space

An experiment is any process that generates well- defined outcomes.

The sample space for an experiment is the set of all experimental outcomes.

An experimental outcome is also called a sample point.

Roll a die 1 3 4 52 6

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An Experiment and Its Sample Space

Experiment

Toss a coin

Inspect a part

Conduct a sales call

Experiment Outcomes

Head, tail

Defective, non-defective

Purchase, no purchase

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Bradley has invested in two stocks, Markley Oil and Collins Mining. Bradley has determined

that thepossible outcomes of these investments three

monthsfrom now are as follows.

Investment Gain or Loss in 3 Months (in $000)

Markley Oil Collins Mining 10 5 0-20

8-2

Bradley Investments

An Experiment and Its Sample Space

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A Counting Rule for Multiple-Step Experiments

If an experiment consists of a sequence of k steps in which there are n1 possible results for the first

step, n2 possible results for the second step, and

so on, then the total number of experimental outcomes is given by:

# outcomes = (n1)(n2) . . . (nk)

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Bradley Investments can be viewed as a two-step

experiment. It involves two stocks, each with a set of

experimental outcomes.Markley Oil: n1 = 4Collins Mining: n2 = 2

Total Number of Experimental Outcomes: n1n2 = (4)(2) = 8

A Counting Rule for Multiple-Step Experiments

Bradley Investments

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Tree Diagram

Gain 5

Gain 10

Lose 20

Even

Markley Oil(Stage 1)

Collins Mining(Stage 2)

ExperimentalOutcomes

(10, 8) Gain $18,000

(10, -2) Gain $8,000

(5, 8) Gain $13,000

(5, -2) Gain $3,000

(0, 8) Gain $8,000

(0, -2) Lose $2,000

(-20, 8) Lose $12,000

(-20, -2) Lose $22,000

Gain 8

Gain 8

Gain 8

Gain 8

Lose 2

Lose 2

Lose 2

Lose 2

Bradley Investments

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Combinations enable us to count the number of experimental outcomes when n objects are to be selected from a set of N objects.

Counting Rule for Combinations

CN

nN

n N nnN

!

!( )!

Number of Combinations of N Objects Taken n at a Time

where: N! = N(N - 1)(N - 2) . . . (2)(1) n! = n(n - 1)(n - 2) . . . (2)(1) 0! = 1

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Number of Permutations of N Objects Taken n at a Time

where: N! = N(N - 1)(N - 2) . . . (2)(1) n! = n(n - 1)(n - 2) . . . (2)(1) 0! = 1

P nN

nN

N nnN

!!

( )!

Counting Rule for Permutations

Permutations enable us to count the number of experimental outcomes when n objects are to be selected from a set of N objects, where the order of selection is important.

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Combinations and Permutations

4 Objects: A B C D

122

24

!2

!4

)!24(

!4

)!(

!42

nN

NP

A B

A C

A D

B C

B D

C D

B A

C A

D A

C B

D B

D C

)!24(!2

!4

)!(!

!42

nNn

NC

64

24

2*2

24

!2!*2

!4

A B

A C

A D C D

B D

B C

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PRACTICETREE DIAGRAMS, COMBINATIONS, AND PERMUTATIONS

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Practice Tree Diagram

A box contains six balls: two green, two blue, and two red.You draw two balls without looking.

How many outcomes are possible?

Draw a tree diagram depicting the possible outcomes.

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Combinations

There are five boxes numbered 1 through 5. You pick two boxes.

How many combinations of boxes are there?

Show the combinations.

)!(!

!

nNn

NC Nn

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Combinations

There are five boxes numbered 1 through 5. You pick two boxes.

How many permutations of boxes are there?

Show the permutations.

)!(

!

nN

NPNn

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