Business and Finance College

Principles of Statistics

Eng. Heba Hamad2008-2009

Slides Prepared bySlides Prepared by

JOHN S. LOUCKSJOHN S. LOUCKSSt. Edward’s UniversitySt. Edward’s University

Slides Prepared bySlides Prepared by

JOHN S. LOUCKSJOHN S. LOUCKSSt. Edward’s UniversitySt. Edward’s University

A second useful counting rule enables 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

Number of Number of CombinationsCombinations of of NN Objects Taken Objects Taken nn at a Time at a Time

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

Example: 4 Colored Balls

• 4 colored balls (R, G, B, Y)

• Pull 2 out of a box at a time

• How many combinations?

Example: 4 Colored Balls

• Formula:– N = 4– n = 2– 4!/[2!(4-2)!] = 4!/[2!2!) = 24/4 = 6

Example:

Exercise :

Number of Permutations of N Objects Taken n at a Time

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

Counting Rule for PermutationsCounting Rule for Permutations

A third useful counting rule enables 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.

Arranging Items Review

• When does order matter (permutation)?– Events in sequence

• When is order not an issue (combination)?– When occurrence is all that matters

The counting rule for Permutations is closely related to the one for Combinations, However there are more permutations than combinations for the same number of objects.

Assigning Probabilities

The three approaches most frequently used are the classical, relative frequency and subjective methods. Regardless of the method used, the probabilities assigned must satisfy two basic requirements

1)(0 iEP For all i

1)()()()( 321 nEPEPEPEP

Assigning Probabilities

Classical MethodClassical Method

Relative Frequency MethodRelative Frequency Method

Subjective MethodSubjective Method

Assigning probabilities based on the assumptionAssigning probabilities based on the assumption of of equally likely outcomesequally likely outcomes

Assigning probabilities based on Assigning probabilities based on experimentationexperimentation or historical dataor historical data

Assigning probabilities based on Assigning probabilities based on judgmentjudgment

Classical Method

If an experiment has n possible outcomes, this method would assign a probability of 1/n to each outcome.

Experiment: Rolling a dieExperiment: Rolling a die

Sample Space: Sample Space: SS = {1, 2, 3, 4, 5, 6} = {1, 2, 3, 4, 5, 6}

Probabilities: Each sample point has aProbabilities: Each sample point has a 1/6 chance of occurring1/6 chance of occurring

ExamplExamplee

Relative Frequency Method

Relative Frequency Method

Number ofNumber ofCars RentedCars Rented

NumberNumberof Daysof Days

0011223344

44 6618181010 22

Lucas Tool Rental would like to assign

probabilities to the number of cars it rents each day. Office records show the following frequencies of daily rentals for the last 40 days.

Example:

Each probability assignment is given bydividing the frequency (number of days) bythe total frequency (total number of days).

Relative Frequency Method

4/40

ProbabilityProbabilityNumber ofNumber of

Cars RentedCars RentedNumberNumberof Daysof Days

0011223344

44 6618181010 224040

.10.10 .15.15 .45.45 .25.25 .05.051.001.00

Stage1 Stage2 Notation for E.O Total Project Comp. times

2

2

2

3

3

3

4

4

4

8

9

10

9

10

11

10

11

12

K- Power & Light Company (KP&L)

6

7

8

6

7

8

6

7

8

(2,6)

(2,7)

(2,8)

(3,6)

(3,7)

(3,8)

(4,6)

(4,7)

(4,8)

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.

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 monthsfrom 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

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

An An eventevent is a collection of sample points.is a collection of sample points.

The The probability of any eventprobability of any event is equal to the sum of is equal to the sum of the probabilities of the sample points in the event.the probabilities of the sample points in the event.

If we can identify all the sample points of anIf we can identify all the sample points of an experiment and assign a probability to each, weexperiment and assign a probability to each, we can compute the probability of an event.can compute the probability of an event.

Events and Their ProbabilitiesEvents and Their Probabilities

Events and Their Probabilities

Event Event MM = Markley Oil Profitable = Markley Oil Profitable

MM = {(10, 8), (10, = {(10, 8), (10, 2), (5, 8), (5, 2), (5, 8), (5, 2)}2)}

PP((MM) = ) = PP(10, 8) + (10, 8) + PP(10, (10, 2) + 2) + PP(5, 8) + (5, 8) + PP(5, (5, 2)2)

= .20 + .08 + .16 + .26= .20 + .08 + .16 + .26

= .70= .70

Events and Their ProbabilitiesEvents and Their Probabilities

Event Event CC = Collins Mining Profitable = Collins Mining Profitable

CC = {(10, 8), (5, 8), (0, 8), ( = {(10, 8), (5, 8), (0, 8), (20, 8)}20, 8)}

PP((CC) = ) = PP(10, 8) + (10, 8) + PP(5, 8) + (5, 8) + PP(0, 8) + (0, 8) + PP((20, 8)20, 8)

= .20 + .16 + .10 + .02= .20 + .16 + .10 + .02

= .48= .48

Assigning Probabilities to Experimental Outcomes

• Using a deck of 52 playing cards, what is the probability of drawing an ace on the first try?

Assigning Probabilities to Experimental Outcomes

• Classical method: 4/52• Relative frequency: conduct an

experiment where you draw one card from deck, then shuffle, and repeat 100 times (e.g. 10 of 100)

• Subjective: based on my experience, I think it is about 1 in 20