1 Slide

Introduction to ProbabilityUncertainty, Probability, Tree Diagrams,

Combinations and Permutations

Chapter 4BA 201

2 Slide

PROBABILITY

3 Slide

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?

4 Slide

Probability

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

Probability values are from 0 to 1.

5 Slide

Probability as a Numerical Measureof the Likelihood of Occurrence

0 10.5Increasing Likelihood of Occurrence

Probability:

The eventis veryunlikelyto occur.

The occurrenceof the event is

just as likely asit is unlikely.

The eventis almostcertain

to occur.

6 Slide

STATISTICAL EXPERIMENTS

7 Slide

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.

8 Slide

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

9 Slide

An Experiment and Its Sample Space

ExperimentToss a coinInspect a partConduct a sales call

Experiment OutcomesHead, tailDefective, non-defectivePurchase, no purchase

10 Slide

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

11 Slide

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)

12 Slide

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

A Counting Rule for Multiple-Step Experiments

Bradley Investments

13 Slide

Tree Diagram

Gain 5

Gain 10

Lose 20Even

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

14 Slide

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

CNn

Nn N nn

N

-!

!( )!

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

15 Slide

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 nNn

NN nn

N

-

! !( )!

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.

16 Slide

Combinations and Permutations

4 Objects: A B C D

12224

!2!4

)!24(!4

)!(!4

2 -

-

nN

NP

A BA CA DB CB DC D

B AC AD AC BD BD C

)!24(!2!4

)!(!!4

2 -

-

nNnNC

6424

2*224

!2!*2!4

A B

A C

A D C D

B D

B C

17 Slide

PRACTICETREE DIAGRAMS, COMBINATIONS, AND PERMUTATIONS

18 Slide

Practice Tree DiagramA 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.

19 Slide

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 -

20 Slide

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 -

21 Slide