Introduction to Probability

•Experiments

•Counting Rules

•Combinations

•Permutations

•Assigning Probabilities

ExperimentsThese are

processes that generate well-

defined outcomes

Experiment Experimental Outcomes

Toss a coin Head, tail

Select a part for inspection

Defective, nondefective

Conduct a sales call Purchase, no purchase

Roll a die 1, 2, 3, 4, 5, 6

Play a football game Win, lose, tie

Probability is a numerical measure of

the likelihood of an event occurring

Probability:

0 1.00.5

The occurrence of the event is just as likely as it is unlikely

Sample Space

TailHead,S

veNondefecti Defective,S

6 ,5 ,4 ,3 ,2 ,1S

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

For a coin toss:

Selecting a part for inspection:

Rolling a die:

Counting Experimental OutcomesTo assign probabilities, we must

first count experimental outcomes. We have 3 useful

counting rules for multiple-step experiments. For example, what

is the number of possible outcomes if we roll the die 4

times?

1. Counting rule for multi-step experiments

2. Counting rule for combinations

3. Counting rule for permutations

Counting Rule for Multi-Step Experiments

If an experiment can be described as a sequence of k steps with n1 possible outcomes on the fist step, n2 possible outcomes on the second step, then the total number of experimental outcomes is given by:

)( . . . ))(( 21 knnn

Example: Bradley Investments

Bradley has invested in two stocks, Markley Oil and Collins Mining. Bradley has determined that the possible outcomes of these investments three months from now are as follows.

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

Markley OilMarkley Oil Collins MiningCollins Mining

1010 55 002020

8822

Bradley Investments can be viewed as atwo-step experiment. It involves two stocks, eachwith a set of experimental outcomes.

Markley Oil:Markley Oil: nn11 = 4 = 4

Collins Mining:Collins Mining: nn22 = 2 = 2Total Number of Total Number of

Experimental Outcomes:Experimental Outcomes: nn11nn22 = (4)(2) = 8 = (4)(2) = 8

A Counting Rule for Multiple-Step Experiments

Tree Diagram

Gain 5Gain 5

Gain 8Gain 8

Gain 8Gain 8

Gain 10Gain 10

Gain 8Gain 8

Gain 8Gain 8

Lose 20Lose 20

Lose 2Lose 2

Lose 2Lose 2

Lose 2Lose 2

Lose 2Lose 2

EvenEven

Markley OilMarkley Oil(Stage 1)(Stage 1)

Collins MiningCollins Mining(Stage 2)(Stage 2)

ExperimentalExperimentalOutcomesOutcomes

(10, 8) (10, 8) Gain $18,000 Gain $18,000

(10, -2) (10, -2) Gain $8,000 Gain $8,000

(5, 8) (5, 8) Gain $13,000 Gain $13,000

(5, -2) (5, -2) Gain $3,000 Gain $3,000

(0, 8) (0, 8) Gain $8,000 Gain $8,000

(0, -2) (0, -2) Lose Lose $2,000$2,000

(-20, 8) (-20, 8) Lose Lose $12,000$12,000

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

Counting Rule for Combinations

)!(!

!

nNn

N

n

NC Nn

)1)(2(. . . )2)(1(!

)1)(2(. . . )2)(1(!

nnnn

NNNN

This rule allows us to count the number of

experimental outcomes when we select n objects

from a (usually larger) set of N objects.

The number of N objects taken n at a time is

where

And by definition 1!0

Example: Quality Control

An inspector randomly selects 2 of 5 parts for inspection. In a group of 5 parts, how many combinations of 2 parts can be selected?

1012

120

)1)(2)(3)(1)(2(

)1)(2)(3)(4)(5(

)!25(!2

!5

2

552

C

Let the parts de designated A, B, C, D, E. Thus we could select:

AB AC AD AE BC BD BE CD CE and DE

Ohio Lottery

Ohio randomly selects 6 integers from a group of 47 to determine the weekly winner. What are your odds of winning if your purchased one ticket?

573,737,10)1)(2)(3)(4)(5)(6(

)42)(43)(44)(45)(46)(47(

)!647(!6

!47

6

47476

C

Counting Rule for Permutations Sometimes the order of

selection matters. This rule allows us to count the number of experimental

outcomes when n objects are to be selected from a set of N objects and the

order of selection matters.

)!(

!!

nN

N

N

NnPNn

Example: Quality Control Again

An inspector randomly selects 2 of 5 parts for inspection. In a group of 5 parts, how many permutations of 2 parts can be selected?

206

120

)1)(2)(3(

)1)(2)(3)(4)(5(

!3

!5

)!25(

!552

P

Again let the parts de designated A, B, C, D, E. Thus we could select:

AB BA AC CA AD DA AE EA BC CB BD DB BE EB CD DC CE EC DE and ED

Basic Requirements for Assigning Probabilities

Let Ei denote the ith experimental outcome and P(Ei) is its probability of occurring. Then:

The sum of the probabilities for all experimental outcomes must be must equal 1. For n experimental outcomes:

iEP i allfor 1)(0

1)( . . . )()( 21 nEPEPEP

Classical Method

This method of assigning probabilities

is indicated if each experimental outcome

is equally likely

nEP i

1)(

Example: Tossing a Die

Experimental Outcome P(Ei)

1 1/6 = .1667

2 1/6 = .1667

3 1/6 = .1667

4 1/6 = .1667

5 1/6 = .1667

6 1/6 = .1667

ΣP(Ei) 1.00

Relative Frequency Method This method is indicated when the data are available to estimate the proportion of the time the experimental outcome will occur if the experiment is repeated a large number of times.

What if experimental outcomes are NOT equally likely. Then the Classical method is out. We must

assign probabilities on the basis of experimentation or

historical data.

Example: Lucas Tool Rental

Relative Frequency Method

Lucas Tool Rental would like to assign probabilities to the number of car polishers it rents each day. Office records show the following frequencies of daily rentals for the last 40 days.

Number ofNumber ofPolishers RentedPolishers Rented

NumberNumberof Daysof Days

0011223344

44 6618181010 22

Each probability assignment is given by dividing the frequency (number of days) by the total frequency (total number of days).

Relative Frequency Method

4/404/404/404/40

ProbabilityProbabilityNumber ofNumber of

Polishers RentedPolishers RentedNumberNumberof Daysof Days

0011223344

44 6618181010 224040

.10.10 .15.15 .45.45 .25.25 .05.051.001.00

Subjective Method When economic conditions and a company’sWhen economic conditions and a company’s circumstances change rapidly it might becircumstances change rapidly it might be inappropriate to assign probabilities based solely oninappropriate to assign probabilities based solely on historical data.historical data. We can use any data available as well as ourWe can use any data available as well as our experience and intuition, but ultimately a probabilityexperience and intuition, but ultimately a probability value should express our value should express our degree of beliefdegree of belief that the that the experimental outcome will occur.experimental outcome will occur.

The best probability estimates often are obtained byThe best probability estimates often are obtained by combining the estimates from the classical or relativecombining the estimates from the classical or relative frequency approach with the subjective estimate.frequency approach with the subjective estimate.

Subjective MethodApplying the subjective method, an analyst

made the following probability assignments.

Exper. OutcomeExper. OutcomeNet Gain Net Gain oror Loss Loss ProbabilityProbability(10, 8)(10, 8)(10, (10, 2)2)(5, 8)(5, 8)(5, (5, 2)2)(0, 8)(0, 8)(0, (0, 2)2)((20, 8)20, 8)((20, 20, 2)2)

$18,000 Gain$18,000 Gain $8,000 Gain$8,000 Gain $13,000 Gain$13,000 Gain $3,000 Gain$3,000 Gain $8,000 Gain$8,000 Gain $2,000 Loss$2,000 Loss $12,000 Loss$12,000 Loss $22,000 Loss$22,000 Loss

.20.20

.08.08

.16.16

.26.26

.10.10

.12.12

.02.02

.06.06