Block 3 Discrete Systems Lesson 11 Discrete Probability ModelsThe world is an uncertain place

Random ProcessRandom happens by chance, uncertain, non-deterministic, stochasticRandom Process a process having an observable outcome which cannot be predicted with certaintyone of several outcomes will occur at random

Uses of ProbabilityMeasures uncertaintyFoundation of inferential statisticsBasis for decision models under uncertaintyA manager observingan uncertainty outcome

Two Approaches to modeling probabilitySample Space and Random EventsUses sets and set theoryRandom Variables and Probability DistributionsDiscrete case algebraicContinuous case - calculus

Sample Spaces and Random EventsLet S = the set of all possible outcomes (events) from a random process. Then S is called the sample space.Let E = a subset of S. Then E is called a random event.Basic problem: Given a random process and the sample space S, what is the probability of the event E occurring - P(E).

Example Sample Space & Random EventsLet S = the set of all outcomes from observing the number of demands on a given day for a particular product. S = {0, 1, 2,n}Let E1 = the random event, there is one demand. Then E1 = {1}Let E2 = the random event of no more than 3 demands. Then E2 = {0, 1, 2, 3}Let E3 = the random event, there are at least 4 demands. Then E3 = {4, 5, , n} or E3 = E2c S x S = the set of outcomes from observing two days of demands = {(0,0), (1,0), (0,1), }

What is a probability P(E)?Let P(E) =Probability of the event E occurring, then0 P(E) 1If P(E) = 0, then event will not occur (impossible event)If P(E) = 1, then the event will occur, i.e. a certain event So the closer P(E) is to 1, then the more likely it is that the event E will occur?

The Sample Space

The collection of all possible outcomes (events) relative to a random process is called the sample space, S whereS = {E1, E2 ... Ek} and P(S) = 1

sampling space

How are probabilities determined?Elementary or basic eventsEmpirical or relative frequencyA priori or equally-likely using counting methodsSubjectively personal judgment or beliefCompound events formed from unions, intersections, and complements of basic eventsLaws of probability

Example - Relative Frequency (empirical) A coin is tossed 2,000 times and heads appear 1,243 times. P(H) = 1243/2000 = .6215The companys Web site has been down 5 days out of the last month (30 days). P(site down) = 5/30 = .166672 out of every 20 units coming off the production line must be sent back for rework.P(rework) = 2/20 = .1

Will you die this year?

Chart1

0.007644

0.000202

0.00011

0.00081

0.000964

0.00129

0.001379

0.001389

0.00177

0.002589

0.003891

0.005643

0.008106

0.012405

0.019102

Male

Age

Probability Male Death

Sheet1

MaleFemaleMaleFemale

ageDeathLifeDeathLifeageDeathLifeDeathLife

expectancyexpectancyexpectancyexpectancy

00.00764474.210.00627579.49260.00134549.720.00051354.47

10.00052873.780.00042178.99270.00132548.790.00053253.49

20.00035772.820.00027378.02280.0013347.850.00055752.52

30.00026871.850.00019677.05290.00135546.910.0005951.55

40.00023270.870.00016876.06300.00138945.980.00062850.58

50.00020269.880.00015275.07310.00142845.040.00067349.61

60.00018668.90.00014274.08320.00148444.10.00072748.65

70.00017167.910.00013573.1330.00156143.170.00079347.68

80.00015166.920.00012872.11340.00165742.240.00086946.72

90.00012765.930.00011971.11350.0017741.310.00095345.76

100.0001164.940.00011370.12360.00189740.380.00104544.8

110.00011963.950.00011869.13370.00204339.450.00114743.85

120.00017762.960.0001468.14380.00220738.530.00125942.9

130.00029761.970.00018467.15390.00238937.620.00138141.95

140.0004660.980.00024466.16400.00258936.710.00151441.01

150.0006460.010.00031265.18410.00280835.80.00165540.07

160.0008159.050.00037564.2420.00304734.90.001839.14

170.00096458.10.00042363.22430.003306340.00194638.21

180.0010957.150.00044762.25440.00358533.120.00209737.28

190.00118956.220.00045361.27450.00389132.230.00226436.36

200.0012955.280.00045660.3460.00421831.360.00244635.44

210.00138654.350.00046459.33470.00455430.490.00263134.52

220.00144353.430.00047158.36480.00489529.630.00281633.61

230.0014552.50.00047957.38490.00524928.770.0030132.71

240.00142151.580.00048856.41500.00564327.920.00322731.8

250.00137950.650.00049955.44

260.00134549.720.00051354.47

270.00132548.790.00053253.49

280.0013347.850.00055752.52

290.00135546.910.0005951.55

300.00138945.980.00062850.58

310.00142845.040.00067349.61

320.00148444.10.00072748.65

330.00156143.170.00079347.68

340.00165742.240.00086946.72

350.0017741.310.00095345.76

360.00189740.380.00104544.8

370.00204339.450.00114743.85

380.00220738.530.00125942.9

390.00238937.620.00138141.95

400.00258936.710.00151441.01

410.00280835.80.00165540.07

420.00304734.90.001839.14

430.003306340.00194638.21

440.00358533.120.00209737.28

450.00389132.230.00226436.36

460.00421831.360.00244635.44

470.00455430.490.00263134.52

480.00489529.630.00281633.61

490.00524928.770.0030132.71

500.00564327.920.00322731.8

510.00607927.070.00347630.9

520.00653826.240.00376330.01

530.00701825.40.00409129.12

540.00753524.580.00446528.24

550.00810623.760.00488427.36

560.00875522.950.00534926.5

570.009522.150.00586125.64

580.01035621.360.00642324.78

590.0113220.580.0070423.94

600.01240519.810.00773223.11

610.01358919.050.00849722.28

620.0148418.310.00931821.47

630.01614917.570.01019220.67

640.01754716.850.01113819.8880.85

650.01910216.150.01219919.09

660.02084715.450.01338418.32

670.02276714.770.01466917.56

680.02487814.10.01605516.82

690.02720113.450.01757116.08

700.02982412.810.01931215.36

710.03271912.190.02126514.66

720.03579511.590.02333313.96

730.039031110.025513.29

740.04251810.420.0278512.62

750.0464999.860.03058211.97

760.0510039.320.03374911.33

770.0558738.790.03725310.71

780.0611048.290.0411110.1

790.0668447.790.0454269.51

800.0732697.310.0503968.94

810.0805726.850.0560988.39

820.0888586.410.0624877.86

830.0982355.990.0696057.35

840.1086945.580.0775526.86

850.1201865.20.0864436.4

860.1326724.850.0963775.96

870.1461374.510.1074275.54

880.1605934.20.119645.14

890.1760743.90.1330354.78

900.1926153.630.1476164.43

910.210243.380.1633764.11

920.2289683.150.1802973.82

930.2487982.930.1983533.55

940.2697172.740.2175093.3

950.2905572.560.2369243.08

960.3110262.410.2563392.88

970.3308172.270.2754692.7

980.3496132.150.2940122.54

990.3670932.040.3116532.39

1000.3854481.930.3303522.25

1010.404721.820.3501732.11

1020.4249561.720.3711841.98

1030.4462041.630.3934551.86

1040.4685141.530.4170621.74

1050.491941.440.4420861.63

1060.5165371.360.4686111.52

1070.5423641.280.4967281.41

1080.5694821.20.5265311.31

1090.5979561.120.5581231.22

1100.6278541.050.591611.13

1110.6592460.980.6271071.05

1120.6922090.920.6647330.97

1130.7268190.850.7046170.89

1140.763160.790.7468940.82

1150.8013180.730.7917080.75

1160.8413840.680.839210.68

1170.8834530.630.8834530.63

1180.9276250.570.9276250.57

1190.9740070.530.9740070.53

Sheet1 (2)

DeathDeath

ageMaleFemaleageMaleFemale

00.0076440.00627500.0076440.006275

10.0005280.00042150.0002020.000152

20.0003570.000273100.000110.000113

30.0002680.000196160.000810.000375

40.0002320.000168170.0009640.000423

50.0002020.000152200.001290.000456

60.0001860.000142250.0013790.000499

70.0001710.000135300.0013890.000628

80.0001510.000128350.001770.000953

90.0001270.000119400.0025890.001514

100.000110.000113450.0038910.002264

110.0001190.000118500.0056430.003227

120.0001770.00014550.0081060.004884

130.0002970.000184600.0124050.007732

140.000460.000244650.0191020.012199

150.000640.000312700.0298240.019312

160.000810.000375750.0464990.030582

170.0009640.000423800.0732690.050396

180.001090.000447850.1201860.086443

190.0011890.000453900.1926150.147616

200.001290.000456

210.0013860.000464

220.0014430.000471

230.001450.000479

240.0014210.000488

250.0013790.000499

260.0013450.000513

270.0013250.000532

280.001330.000557

290.0013550.00059

300.0013890.000628

310.0014280.000673

320.0014840.000727

330.0015610.000793

340.0016570.000869

350.001770.000953

360.0018970.001045

370.0020430.001147

380.0022070.001259

390.0023890.001381

400.0025890.001514

410.0028080.001655

420.0030470.0018

430.0033060.001946

440.0035850.002097

450.0038910.002264

460.0042180.002446

470.0045540.002631

480.0048950.002816

490.0052490.00301

500.0056430.003227

510.0060790.003476

520.0065380.003763

530.0070180.004091

540.0075350.004465

550.0081060.004884

560.0087550.005349

570.00950.005861

580.0103560.006423

590.011320.00704

600.0124050.007732

610.0135890.008497

620.014840.009318

630.0161490.010192

640.0175470.011138

650.0191020.012199

660.0208470.013384

670.0227670.014669

680.0248780.016055

690.0272010.017571

700.0298240.019312

710.0327190.021265

720.0357950.023333

730.0390310.0255

740.0425180.02785

750.0464990.030582

760.0510030.033749

770.0558730.037253

780.0611040.04111

790.0668440.045426

800.0732690.050396

810.0805720.056098

820.0888580.062487

830.0982350.069605

840.1086940.077552

850.1201860.086443

860.1326720.096377

870.1461370.107427

880.1605930.11964

890.1760740.133035

900.1926150.147616

910.210240.163376

920.2289680.180297

930.2487980.198353

940.2697170.217509

950.2905570.236924

960.3110260.256339

970.3308170.275469

980.3496130.294012

990.3670930.311653

1000.3854480.330352

1010.404720.350173

1020.4249560.371184

1030.4462040.393455

1040.4685140.417062

1050.491940.442086

1060.5165370.468611

1070.5423640.496728

1080.5694820.526531

1090.5979560.558123

1100.6278540.59161

1110.6592460.627107

1120.6922090.664733

1130.7268190.704617

1140.763160.746894

1150.8013180.791708

1160.8413840.83921

1170.8834530.883453

1180.9276250.927625

1190.9740070.974007

Sheet1 (2)

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Male

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Sheet2

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Probability Male Death

Sheet3

Example - A priori (equally-likely outcomes)A pair of fair dice are tossed. S x S = {(1, 1),(1, 2),(1, 3),(1, 4),(1, 5),(1, 6), (2, 1),(2, 2),(2, 3),(2, 4),(2, 5),(2, 6),(3, 1),(3, 2),(3, 3),(3, 4),(3, 5),(3, 6),(4, 1),(4, 2),(4, 3),(4, 4),(4, 5),(4, 6),(5, 1),(5, 2),(5, 3),(5, 4),(5, 5),(5, 6),(6, 1),(6, 2),(6, 3),(6, 4),(6, 5),(6, 6)}P(a seven) = 6/36 = .1667A supply bin contains 144 bolts to be used by the manufacturing cell in the assembly of an automotive door panel. The supplier of the bolts has indicated that the shipment contains 7 defective bolts. Let E = the event, a defective bolt is selectedP(E) = 7/144 = .04861.

A priori (equally-likely outcomes) versus Relative Frequency (empirical) A prior (knowable independently of experience):

n(A) = the number of ways in which event A can occurn(S) = total number of outcomes from the random process

Relative frequency:

where n(E) = number of times event E occurs in n trials

Examples Subjective Probability8 out of 10 leading economists believe that the gross national product (GNP) will grow by at least 3% this year.P(GNP .03) = 8/10 = .8Bigg Bosse, the CEO for a major corporation, consults his marketing staff. Together they make the assessment that there is a 50-50 chance that sales will increase next year.The House of Congress majority leader, after consulting with his staff, determines that there is only 25 percent chance that an important tax bill will be passed.

Computing Probabilities for Compound EventsFinding the probability of the union, intersection, and complements of events

Mutually Exclusive EventsP(Ac ) = 1 - P(A)

P( A B) = P(A) + P(B) if A and B are mutually exclusive

P(A B) = P() = 0 if A and B are mutually exclusiveABNote that A and B are not mutually exclusive.

More Mutually Exclusive EventsRandom process: draw a card at random from an ordinary deck of 52 playing cardsLet A = the event, an ace is drawnLet B = the event, a king is drawn P(A) = 1/13 and P(B) = 1/13Then P( A B) = P(A) + P(B) = 1/13 + 1/13 = 2/13P(A B) = 0P(A) = 1 1/13 = 12/13; P(B) = 12/13P(A B) = P( A B) = ? the event is not an ace or not king

Its not me!

The Addition FormulaP(A B) = P(A) + P(B) - P(A B)

ABA B

More of the Addition FormulaRandom process: draw a card at random from an ordinary deck of 52 playing cardsLet A = the event, draw a spadeP(A) = 13/52 =1/4let B = the event draw an aceP(B) = 4/52 = 1/13P(A B) = 1/52P(A B) = P(A) + P(B) - P(A B) = 1/4 + 1/13 1/52 = 13/52 + 4/52 1/52 = 16/52 = .3077

112336

The Multiplication ruleIndependent EventsTwo events, A and B are independent if the P(A) is not affected by the event B having occurred (and vice-versa).

If A and B are independent, thenP(A B) = P(A) P(B)

An IndependenceeventNote that A and B are independentif A and B are independent.

Proof by ExampleLet E1 = the event, a three or four is rolled on the toss of a single fair die, P(E1) = 2/6 E2 = the event, a head is tossed from a fair coin, P(E2) = 1/2

then D x C = {(1,H), (1,T), (2,H), (2,T), (3,H), (3,T), (4,H), (4,T), (5,H), (5,T), (6,H), (6,T)}; P(E1 E2) = 2/12

Mult. rule: P(E1 E2) = P(E1) P(E2) = (2/6) (1/2) = 2/12

Another ExampleLet A = the event, prototype A fails heat stress test B = the event prototype B fails a vibration testGiven P(A) = .1 ; P(B) = .3Find P(A B) = ? and P(A B) = ?

Assuming independence:P(A B) = P(A) P(B) = (.1)(.3) = .03P(A B) = P(A) + P(B) - P(A B) = .1 + .3 - .03 = .37

P(A B) = P(A B) = P(A)P(B) = (.9)(.7) = .63

Glee Laundry DetergentEach box of powered laundry detergent coming off of the final assembly line is subject to an automatic weighing to insure that the weight of the contents falls within specification. Each box is then visually inspected by a quality assurance technician to insure that is it properly sealed.

GleeI have been rejected.

More GleeIf three percent of the boxes fall outside the weight specifications and five percent are not properly sealed, what is the probability that a box will be rejected after final assembly?Let A = the event, a box does not meet the weight specification; P(A) = .03Let B = the event, a box is not properly sealed; P(B) = .05

P(A B) = P(A) + P(B) P(A)P(B) = .03 + .05 - .0015 = .0785

A Reliability ProblemAn assembly is composed of 3 components as shown below. If A is the event, component A does not fail, B is the event, component B does not fail, and C is the event, component C does not fail, find the reliability of the assembly where P(A) = .8, P(B) = .9, and P(C) = .8. Assume independence among the components. P(S) = P[ (A B) C] = P(A B) + P(C) P(A B C) = P(A) P(B) + P(C) P(A)P(B)P(C) = (.8)(.9) + .8 (.8)(.9)(.8)= .72 + .8 - .576 = .944

Next random variables and their probability distributions!Variables that are random; what will they think of next?

Discrete Random VariablesA random variable (RV) is a variable which takes on numerical values in accordance with some probability distribution. Random variables may be either continuous (taking on real numbers) or discrete (usually taking on non-negative integer values). The probability distribution which assigns probabilities to each value of a discrete random variable can be described in terms of a probability mass function (PMF), p(x) in the discrete case.

Random Variables - ExamplesY = a discrete random variable, the number of machines breaking down each shift X = a discrete random variable, the monthly demand for a replacement partZ = a discrete random variable, the number of hurricanes striking the Gulf Coast each yearXi = a discrete random variable, the number of products sold in month i

The PMFThe Probability Mass Function (PMF), p(x), is defined as p(x) = Pr(X = x}

and has two properties:By convention, capital letters represent the random variable while the corresponding small letters denote particular values the random variable may assume.

A Probability DistributionLet X = a RV, the number of customer complaints received each dayThe Probability Mass Function (PMF) is

The CDFThe cumulative distribution function (CDF), F(x) is

defined where

Rolling the diceX = the outcome from rolling a pair of dice number of ways

2(1,1)13(1,2) , (2,1)24(1,3) , (2,2) , (3,1)35(1,4) , (2,3) , (3,2) , (4,1)46(1,5) , (2,4) , (3,3) , (4,2) , (5,1)57(1,6) , (2,5) , (3,4) , (4,3) , (5,2) , (6,1)68(2,6) , (3,5) , (4,4) , (5,3) , (6,2)59(3,6) , (4,5) , (5,4) , (6,3)410(4,6) , (5,5) , (6,4)311(5,6) , (6,5)212(6,6)1Total36

An ExampleLet X = a random variable, the sum resulting from the toss of two fair dice; X = 2, 3, , 12

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) = S

Probability Histogram for the Random Variable X

Expected Value or MeanThe expected value of a random variable (or equivalently the mean of the probability distribution) is defined asDont you get it? The expected value is just a weighted average of the values that the random variable takes on where the probabilities are the weights.

Example Expected ValueE[X] = 0 (.1) + 1 (.3) + 2 (.5) + 3 (.1) = 1.6I expected this to have a little value!

Example Expected ValueDice example:

= 2(1/36) + 3(2/36) + 4(3/36) + 5(4/36) + 6(5/36) + 7(6/36) + 8(5/36) + 9(4/36) + 10(3/36) + 11(2/36) + 12(1/36) = (1/36) (2 + 6 + 12 + 20 + 30 + 42 + 40 + 36 + 30 + 22 + 12) = (252/36) = 7

Yet Another Discrete DistributionLet X = a RV, the number of customers per dayF(20) = (20)(21) / 420 = 1

More of yet another discrete distributionLet X = a RV, the number of customers per day

Pr{X = 15} = p(15) = 15/210 = .0714Pr{X 15} = F(15) = (15)(16)/420 = .5714Pr{10 < X 15} = F(15) F(10) = .5714 (10)(11)/420 = .5714 - .2619 = .3095

So ends our discussion on Discrete ProbabilitiesComing soon to a classroom near you discrete optimization modelsA most enjoyable lesson.

*

Examples of random processesManufacturingProduct Demand will I be able to sell everything I make? Will I have too much inventory?Product reliability when will this product fail? Will it still be under warranty?Product Quality How many rejects will I have during production?Lead times How many weeks before I receive my supplier shipment?Service IndustryCustomers waiting How long will my customers wait for service? How long will the line be?Delivery times How long will it take to deliver this shipment?Parts Inventory Will I have the necessary repair parts on hand?GovernmentHow much will be paid out this year in social security benefits?What will be revenue from personal and business income taxes?Will there be any natural disasters that the state will assume liability for? If so, how much?*A failure can be defined as a random event.*The complement to the failure event is the event that a failure does not occur. An event and its complement are always mutually exclusive. Why?

*The union of two events is itself an event. This event will occur if either or both of the original events occur. Why is probability of the intersection subtracted out in the above formula?**Some really good examples.**