block 3 discrete systems lesson 11 –discrete probability models the world is an uncertain place
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Block 3 Discrete Systems Lesson 11 Discrete Probability ModelsThe world is an uncertain place
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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
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Uses of ProbabilityMeasures uncertaintyFoundation of inferential statisticsBasis for decision models under uncertaintyA manager observingan uncertainty outcome
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Two Approaches to modeling probabilitySample Space and Random EventsUses sets and set theoryRandom Variables and Probability DistributionsDiscrete case algebraicContinuous case - calculus
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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).
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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), }
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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?
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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
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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
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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
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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
Female
Sheet2
0
0
0
0
0
0
0
0
0
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Male
Age
Probability Male Death
Sheet3
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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.
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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
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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.
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Computing Probabilities for Compound EventsFinding the probability of the union, intersection, and complements of events
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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.
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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!
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The Addition FormulaP(A B) = P(A) + P(B) - P(A B)
ABA B
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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
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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.
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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
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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
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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.
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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
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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
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Next random variables and their probability distributions!Variables that are random; what will they think of next?
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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.
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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
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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.
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A Probability DistributionLet X = a RV, the number of customer complaints received each dayThe Probability Mass Function (PMF) is
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The CDFThe cumulative distribution function (CDF), F(x) is
defined where
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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
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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
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Probability Histogram for the Random Variable X
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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.
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Example Expected ValueE[X] = 0 (.1) + 1 (.3) + 2 (.5) + 3 (.1) = 1.6I expected this to have a little value!
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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
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Yet Another Discrete DistributionLet X = a RV, the number of customers per dayF(20) = (20)(21) / 420 = 1
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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
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So ends our discussion on Discrete ProbabilitiesComing soon to a classroom near you discrete optimization modelsA most enjoyable lesson.
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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.**
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