DMDch02

Instructor’s Manual

Chapter 2

17

Chapter 2 I

Chapter Outline

2.1 Outcomes, Outcomes, Probabiliti Probabilities es and Events • The ollo!in" concepts are introduced#  $andom e%periment Outcome o an e%periment  Mutuall& e%clusive and collectivel& e%haustive outcomes   Probabilit& o an outcome Event  2.2 The 'a!s o Probabili Probabilit& t& $evie! o basic probabilit& rules includin" addition rules, conditional probabilit&, and product la!s, and an introduction o the concept o independence. 2.( )or*in" )or*in" !ith Probabilities and Probabilit& Tables Tables $evie! o decision tree anal&sis in the conte%t o conditional probabilities. 2.+ $andom a ariables riables 2.- iscrete iscrete Probabilit Probabilit& & istributi istributions ons 2./ The 0inomial 0inomial istributi istribution on einition o a binomial random variable and its parameters. Introduction o the combinatorial ormula or the computation co mputation o a binomial probabilit&. probabilit&. 2.7 ummar& Measures o Probabilit& Probabilit& istributions istributions ei eini niti tion on o mean mean or e%pec e%pecte ted d valu value, e, vari varian ance ce,, and and stan standa dard rd devi deviat atio ion. n. erivation o mean and standard deviation o a binomial random variable. 2. 'inear 3unctions o a $andom ari ariable able Evalua Evaluatio tion n o mean, mean, varian variance, ce, and standa standard rd deviat deviation ion o a random random variab variable le e%pressed as a linear unction o another random rando m variable 45 6 a 8 b9. 2.: Covariance Covariance and Correlati Correlation on

Manual to accompan& Data, Models & Decisions: The Fundamentals of Management Science b& 0ertsimas and 3reund. Cop&ri"ht 2;;;, outh<)estern Colle"e Publishin". Prepared b& Manuel =une>, Chapman ?niversit&.


Chapter 2

1

2.1;

@oint Probabilit& istributions and Independence einition o the Aoint probabilit& distribution o t!o random variables. $evie! o independence o random variables under this conte%t. 3ormula or the e%pectation o the product o t!o independent random variables. Independence and the relation to covariance and correlation.

2.11

ums o T!o $andom ariables E%pectation, variance, and standard deviation o the linear combination o t!o random variables.

2.12

ome Bdvanced Methods in Probabilit& iscussion o 0a&es theorem, la! o total probabilit&, and alternate ormula or the variance.

2.1(

ummar&

II

Teaching Tips

1.

It is recommendable to produce several e%amples stressin" the dierence bet!een the classical approach to computin" probabilities and the relative reDuenc& 4empirical9 approach. Man& students have little intuition concernin" the dierence bet!een both o them. This can be done b& en"a"in" the students in a Duic* classroom surve& o a "iven characteristic o the students, and then contrastin" the probabilities rom the surve& to the classical probabilities rom a binomial variable associated to this characteristic.

2.

It is "ood idea to remind students that the e%pectation should be interpreted in a lon"
(.

To help understand the meanin" o the variance, the instructor ma& !ant to introduce the coeicient o variation 4σµ9 as a relative measure o the variabilit& !ithin a population and its possible use to compare populations o ver& dierent nature.

III

Answers to Chapter Exercises

2.1 4a9 P4F6-9 6 P41,+9 8 P42,(9 8 P42,+9 8 P4(,29 8 P4(,(9 8 P4(,+9 8 P4+,19 8 P4+,29 8 P4+,(9 8 P4+,+9 6 1;1/. 4b9 P4F6- G 3irst is (9 6 P4F6- and 3irst is (9P43irst is (9 6 4(1/94+1/96(+. 4c9 P4F6- G Bt least ( in one die9 6 P4F6- and Bt least ( in one die9P4Bt least ( in one die9 6 4-1/9471/96-7.



Chapter 2

1:

2.2 4a9 B total o  outcomes# HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT. 4b9 P4HHT9 6 1. 4c9 P43irst t!o tosses are heads9 6 2. 4d9 P4T!o heads in a ro!9 6 (. 2.( 4a9 P4Bt least + &ears e%perience9 6 P4+ &ears e%perience9 8 P4- or more &ears e%perience9 6 1-1;; 8 (-1;; 6 -;1;;. 4b9 P4Bt least + &ears e%perience G Bt least ( &ears e%perience9 6 P4Bt least + &ears e%perience9P4Bt least ( &ears e%perience9 6 4-;1;;94;1;;9 6 -;;. 2.+ 4a9 P4Oil at all three places9 6 4;.7;9%4;.-9%4;.;9 6 ;.+7/. 4b9 P4=o oil at an& site9 6 4;.(;9%4;.1-9%4;.2;9 6 ;.;;:. 2.4a9 P4unn& on )ednesda&9 6 P4unn& on Tuesda& and sunn& on )ednesda&9 8 P4Cloud& on Tuesda& and sunn& on )ednesda&9 6 4;.(9%4;./;9 8 4;.79%4;.(9 6 ;.1 8 ;.21 6 ;.(:. 4b9 P4unn& on Tuesda& and )ednesda&9 6 4;.(;9%4;./;9 6 ;.1. 2./ 4a9 Bccordin" to the table belo! the prime time slot !ould &ield the hi"hest e%pected net proit 4112,;;;9. Time Slot Morning Afternoon Prime Time &ate E'ening

Ad Cost Estimated Viewers $120000 1000000 $200000 1#00000 $"00000 #200000 $1%0000 00000

Estimated Earnings $1!0000 $20000 $%12000 $12000

Expected Net Profit $"0000 $000 $112000 ($22000

4b9 The compan& should bu& the mornin" and the prime time slots, !ith a total e%pected contribution to earnin"s o 1-2,;;;. 2.7

P4'on"
2.

Consider the decision tree belo!. )e assume that the bo%es are labeled B, 0, and C. )ithout loss o "eneralit&, !e assume that the initiall& chosen bo% is B. Hence, there are our possible scenarios, each !ith the same probabilit&# the pri>e is in B and the host sho!s 0, the pri>e is in B and the host sho!s C, the pri>e is in 0 and the host must sho! C, or the pri>e is in C and the host must sho! 0. Thereore,



Chapter 2

2;

the host sho!s 0 !ith probabilit& 2+ 6 ;.-;, and it sho!s C !ith probabilit& 2+ 6 ;.-;. Bccordin" to this decision tree, &ou should al!a&s chan"e &our pic*.

2.:

'et H denote the person has HI, $ the test is positive, and = the test is ne"ative. Then P4H9 6 --;;;;2-;;;;;;; 6 ;.;;22, P4$ G H9 6 ;.::, and P4= G not H9 6 ;.::. 4a9 )e !ant to compute P4H G $9. =otice that P4$9 6 P4$ G H9%P4H9 8 P4$ G not H9%P4not H9 6 ;.;;217 8 ;.;;::7 6 ;.;121-/. Thereore, P4H G $9 6 P4$ G H9%P4H9P4$9 6 ;.;;217;.;121-/ 6 ;.1. 4b9 'et  denote that the person is a dru" user. )e !ant to compute P4H G $ and 9. =otice that P4H and 9 6 27-;;;2-;;;;;;; 6 ;.;;11 and P44not H9 and 9 6 41;;;;;;; < 27-;;;92-;;;;;;; 6 ;.;(:. )e assume that P4$ G H and 9 6 P4$ G H9 6 ;.:: and P4= G 4not H9 and 9 6 ;.::. Then P4$ and 9 6 4;.::9%4;.;;119 8 4;.;19%4;.;(:9 6 ;.;;1+7. Thereore, P4H G $ and 9 6 P4H and $ and 9P4$ and 9 6 P4$ G H and 9%P4H and 9P4$ and 9 6 4;.::9%4;.;;1194;.;;1+79 6 ;.7+.

2.1;

'et 1 denote the lamp comes rom irst shipment, 2 denote the lamp comes rom second shipment, and  denote the lamp is deective. )e !ant to compare P41 G 9 to P42 G 9. )e have P419 6 1;;1-; 6 2(, P429 6 -;1-; 6 1(, P4 G 19 6 ;.;+, and P4 G 29 6 ;.;/. 3irst, P49 6 P4 G 19% P419 8 P4 G 29%



Chapter 2

21

P429 6 71-;. Thereore, P41 G 9 6 P4 G 19% P419P49 6 1+, and P42 G 9 6 P4 G 29% P429P49 6 /1+. In conclusion, the lamp is more li*el& to come rom the irst shipment. 2.11

'et T0 denote the person is inected !ith the T0 bacteria, and let  denote the person has the disease. )e have P4T09 6 1( and P4 G T09 6 ;.1;. 4a9 P49 6 P4 G T09% P4T09 8 P4 G not T09%P4not T09 6 ;.1;%41(9 8 ;%42(9 6 1(;. 4b9 P4T0 G not 9 6 P4not  G T09%P4T09P4not 9 6 :2:.

2.12

'et  be the portolios return. 4a9 ;.;7 8 ;.1- 8 ;.2( 8 ;.2- 8 ;.1- 8 ;.12 8 ;.;( 6 1. 4b9 P4 F6 12J9 6 ;.2- 8 ;.1- 8 ;.12 8 ;.;( 6 ;.--. 4c9 E49 6 11.7+J. 4d9 B$49 6 2.2:2+K1;<+ and σ49 6 1.-1J.

2.1(

'et  be the micro!ave ovens sold per !ee*. 4a9 P41 L6  L6 (9 6 ;.;7 8 ;.22 8 ;.2: 6 ;.-. 4b9 E49 6 2.:, B$49 6 1./, and σ49 6 1.(.

2.1+ 4a9 The distributions are "iven belo!. Time Tas) A 1 2

P*Time+ 0/"0 0/!0

Time Tas) , 1 2 #

P*Time+ 0/20 0/% 0/22

Time -o. 1 2 #

P*Time+ 0/0 0/1 0/22

4b9 E4Time B9 6 1./, E4Time 09 6 2.;2, and E4Time Aob9 6 2.1- σ4Time B9 6 ;.+:, σ4Time 09 6 ;./-, and σ4Time Aob9 6 ;.-2. 4c9 The distribution is "iven belo!. E4Cost9 6 +,1/; and σ4Cost9 6 :-.7-. Cost $2200 $#"00 $#!00 $""00 $%000 $%00

2.1-

P*Cost+ 0/0 0/1# 0/2 0/#1 0/0! 0/1!

'et  be the number o courses per !ee*. 4a9 ;.;- 8 ;.1- 8 ;.2- 8 ;.2- 8 ;.1- 8 ;.1; 8 ;.;- 6 1. 4b9 P46;9 6 ;.;-. 4c9 P4 F6 (9 6 ;.2- 8 ;.1- 8 ;.1; 8 ;.;- 6 ;.--. 4d9 P4 F6 19 6 1 < P4 6 ;9 6 1 < ;.;- 6 ;.:-.



Chapter 2

22

4e9 and 49 'et $ denote the net revenue. The ollo!in" table summari>es the available options. EN$ 6 -;,-;; and σ4$9 6 22,/;;. The best option is to add one more instructor 4mar"inal increase o ;; in net revenues9.

 P*+ ∆ *1 instr/+ ∆ *2 instr/+ ∆ *# instr/+ $0 0/0% ($2%00 ($%000 ($%00 $20000 0/1% ($2%00 ($%000 ($%00 $"0000 0/2% ($2%00 ($%000 ($%00 $!0000 0/2% ($2%00 ($%000 ($%00 $!000 0/1% $%00 $!000 $#%00 $000 0/10 $%00 $1000 $1"%00 $000 0/0% $%00 $1000 $2%%00 Expectation $00 ($%0 ($2000

2.1/

'et  be the number o boats constructed each month and let C be the monthl& cost o the operation. =otice that C 6 +;; 8 (;;;;. 4a9 E49 6 + and σ49 6 1.(. 4b9 E4C9 6 +;;%+ 8 (;;;; 6 +:,2;; and σ4C9 6 +;;%41.(9 6 /,2+;. 4c9 E4C9 6 +:2;; 8 2(;;; 6 72,2;; and σ4C9 6 /,2+;. 4d9 E4C9 6 +:2;; 8 +%47;;;<+;;9 6 -,;;; and σ4C967;;;41.(9 6 :,1;;.

2.17

'et M be the milea"e o a customer. 4a9 1(; 8 ;.2 M 6 1:- implies M 6 (2-. 0ill drove (;; 8 (2- 6 /2- miles. 4b9 EC$ rate is better i and onl& i M F /2-. P4M F /2-9 6 ;.1( 8 ;.;: 8 ;.; 6 ;.(;. 4c9 ince $B is less e%pensive, that means M L6 /2-. 3rom the table belo!, it ollo!s that E4Cost9 6 1-1. Miles Cost P*Cost+ P*Cost 3 M 45 !2%+ 200 1#0 0/0 0/10 #00 1#0 0/1 0/2 "00 1%0 0/2# 0/## %00 10 0/1" 0/20 !00 10 0/0 0/10 Total 0/0 1/00

Cost x P 1# #% " #" 1 1%1

2.1

The standard deviation is σ409 6 4-:9 % σ4B9 6 1; de"rees Celsius.

2.1:

'et T be the emplo&ees total !ee*l& salar& and let  be the overtime per !ee*. =otice that T 6 +;%12 8 1% 6 +; 8 1. E4T9 6 +; 8 1%1- 6 7-;, σ4T9 6 1%+ 6 72, and B$4T9 6 -1+.

2.2;

'et B, 0, and C be the dail& production rate o the Bndover, 0edord, and Concord plants, respectivel&. =otice that  6 B 8 0 8 C. 4a9 E49 6 E4B9 8 E409 8 E4C9 6 :1 8 /7 8 /: 6 227.



Chapter 2

2(

4b9 B$49 6 B$4B9 8 B$409 8 B$4C9 6 +7.+7 and σ49 6 /.:. 2.21

'et  denote the si>e o the sales orce and let $ denote the &earl& sales revenue. 4a9 E49 6 2/.2-, B$49 6 (:.7, and σ49 6 /.(. 4b9 E4$9 6 2., B$4$9 61.(-, and σ4$9 6 1.1/. 4c9 CO4, $9 6 /. and CO$$4, $9 6 ;.:(.

2.22

'et ?1 be the umbrellas sold at the department store, ?2 be the umbrellas sold at the outlet, and  be the total sales revenue. Then  6 17?1 8 :?2. E49 6 17%41+7.9 8 :%4/(.29 6 (,;1, B$49 6 2:B$4?19 8 1B$4?29 8 2%17%:%-1%(7%CO$$4?1, ?29 6 12//77( and σ49 6 1,12-.-1.

2.2(

CO$$4, 59 6 <417.7947%/.29 6 <;.+1.

2.2+

'et $ 6 ;.- 8 ;.-5 represent the return. 4a9 E4$9 6 4;.-94;.1-9 8 4;.-94;.29 6 ;.17-, B$4$9 6 ;.2-B$49 8 ;.2-B$459 8 24;.-94;.-94;.;-94;.;/9CO$$4, 59 6 ;.;;2, and σ4$9 6 ;.;+-. 4b9 The e%pected return is al!a&s the same E4$9 6 ;.17-. ee table belo! or variance and standard deviation. C6*78+ (0/!0 (0/#0 0/00 0/#0 0/!0

VA*+ 0/000! 0/0011 0/001% 0/0020 0/002"

σ*+

0/02%0 0/0#2 0/0#1 0/0""" 0/0"2

4c9 and 4d9 ee "raph belo!.

Return Plots 0/00"0 0/00#0

. D . 0/0020 S

C65(0/! C65(0/#

0/0010

C650

0/0000

C650/#

0/1"

0/1%

0/1!

0/1

0/1$

0/1

0/20

0/21

C650/!

Mean



Chapter 2

2+

2.2-

'et B denote the inspector accepts a microchip and  denote a microchip is deective. 4a9 4;.:-91; 6 ;./. 4b9 P4B9 6 P4B G 9P49 8 P4B G not 9P4not 9 6 4;.194;.;-9 8 14;.:-9 6 ;.:--. 4c9 P4Bccepts : out o 1;9 6 1;4;.:-9:4;.;-9 6 ;.(2. 4d9 P4not  G B9 6 ;.:-;.:-- 6 ;.::-. 4e9 P4no deects G accepts all 1;9 6 4;.::91; 6 ;.:-.

2.2/

'et  be the number o &ears that both movements a"ree. 4a9 P4B"ree9 6 4;./94;.:9 8 4;.+94;.19 6 ;./. 4b9 4;./92; 6 ;.;;;;(. 4c9 ?sin" a binomial distribution !ith n 6 2; and p 6 ;./, P4 F6 1-9 6 ;.12-. 4d9 ?sin" same distribution, P4 F6 179 6 ;.;1-. 4e9 Bccordin" to 4d9, i the theor& is alse, then there is a ::J probabilit& that the number o &ears that both movements a"ree in a 2;<&ear period is less than 17, that is, it is an unusual event to have  F6 17. o that, i in several random samples o 2; &ears each, the event  F6 17 occurs ver& reDuentl&, then there is indication that the theor& mi"ht be true. ?sin" h&pothesis testin" can ma*e a more ormal statement, !hich at this moment is be&ond the material covered.

2.27

'et  be the number o persons 4out o 119 !ho sho! up.  is binomial !ith parameters n 6 11 and p 6 ;.. 4a9 P4L6-9 6 ;.;12. 4b9 P4 6 1;9 6 114;.91;4;.29 6;.2(/. 4c9 E49 6 114;.9 6 .. E4Proit9 6 12;;E49 < (;;;P46119 6 1;,(;2. 4d9 12;;4;.91; 6 :,/;;. 4e9 5es, because i one person sho!s up, then it is ver& li*el& that hisher companions !ill also sho! up. Thereore, the event a person sho!s up is not independent o the event the ne%t person sho!s up.

2.2 4a9 -;;4;.1-9 6 7-. 4b9 7.:. 2.2: 4a9 P4E%actl& one9 6 :4;.;/94;.:+9 6 ;.((. P4more than one9 6 1 < P4e%actl& one9 < P4none o them9 6 1 < ;.(( < ;.-7 6 ;.1;. 4b9 P4E%actl& one9 6 /4;.1294;.9- 6 ;.(. P4more than one9 6 1 < P4e%actl& one9 < P4none o them9 6 1 < ;.( < ;.+/ 6 ;.1/. 4c9 4;.:+9(4;.92 6 ;./+ 2.(; 4a9 P4Bt least +9 6 ;.;/ 8 ;.;1 8 ;.;;1 6 ;.;71. 4b9 P4Bt most 29 6 ;.11 8 ;.(;( 8 ;.(2+ 6 ;.7+-.



2.(1

Chapter 2

2-

4;.1-9+ 6 ;.;;;-.

2.(2 4a9 The preerence o a "iven customer is independent o the preerence o another customer and the probabilit& that a customer preer thin crust pi>>a is the same or all / customers. 4b9 The distribution is in the table belo!. 7 0 1 2 # " % !

P*7+ 0 0/002 0/01% 0/02 0/2"! 0/## 0/2!2

4c9 P4 6 (9 6 ;.;2 4d9 E49 6 /4;.9 6 +. and σ49 6 ;.:. 2.((

'et  be the number o times the 3und !ill increase over the ne%t 12 months.  is 0inomial !ith n 6 12 and p 6 ;./-. 4a9 P4 6 79 6 ;.2;+. 4b9 E49 6 124;./-9 6 7.. E%pected chan"e 6 E4- < +412<99 6 :E49 < + 6 :47.9 <+ 6 22.2J.

2.(+

'et  be the number o customers 4out o 1;9 !ho are reDuent BTM users.  is 0inomial !ith n 6 1; and p 6 ;.+. 4a9 P4 F6 +9 6 ;./2. 4b9 P4 L6 /9 6 ;.:+-. 4c9 P4+ L6  L6 /9 6 ;.-/(.

2.(-

'et  be the number o clubs 4out o -9 !hich !ill accept @im.  is 0inomial !ith n 6 - and p 6 ;./-. 4a9 P4 6 (9 6 ;.((/+. 4b9 P4 F6 (9 6 P46(9 8 P46+9 8 P46-9 6 ;.((/+ 8 ;.(12+ 8 ;.11/;( 6 ;.7/-. 4c9 1 < 4;.(-9( 6 ;.:/.



Chapter 2

2/

2.(/

'et  be the number o passen"ers 4out o /9 !ho !ill become ill.  is 0inomial !ith parameters n 6 / and p 6 ;.12. P4 L6 29 6 P46;9 8 P4619 8 P4629 6 ;.+/ 8 ;.( 8 ;.1( 6 ;.:7.

2.(7

'et  be the number o devices 4out o -9 !hich !ill sound an alarm !hen no intruder is present.  is 0inomial !ith parameters n 6 - and p 6 ;.1. P4 F6 29 6 P4629 8 P46(9 8 P46+9 8 P46-9 6 ;.;7( 8 ;.;; 8 ;.;;; 8 ;.;;; 6 ;.;1.

2.(

'et  be the number o customers 4out o 1/9 !ho come rom houses in !hich "as is used or residential heatin".  is 0inomial !ith parameters n 6 1/ and p 6 ;.:. P4 F6 129 6 P46129 8 P461(9 8 P461+9 8 P461-9 8 P461/9 6 ;.;-1 8 ;.1+2 8 ;.27- 8 ;.(2: 8 ;.1- 6 ;.:2.

IV

Answers to Chapter Cases

ARIZONA INSTRUMENTATION INC! AN" T#E ECONOMIC "EVE$O%MENT &OAR" O' SIN(A%ORE Consider the decision tree on the ne%t pa"e. Bccordin" to this tree, the optimal decision strate"& is to implement the BE plan, !ith an EM o 7-.- millions. To compute the probabilities o avorable events under the E0BII plan, !e did the ollo!in"# • P4+ avorable events9 6 4;.7;9%4;./;9%4;.-;9%4;.:;9 6 ;.1:, avorable events9 6 4;.7;9%4;./;9%4;.-;9%4;.1;9 8 • P4( 4;.7;9%4;./;9%4;.-;9%4;.:;9 8 4;.7;9%4;.+;9%4;.-;9%4;.:;9 8 4;.(;9%4;./;9%4;.-;9%4;.:;9 6 ;.+17, • P42 or less avorable events9 6 1 < ;.1: < ;.+17 6 ;.(:+. imilarl&, !e ound that under the BE plan# • P4+ avorable events9 6 ;.1(:/, • P4( avorable events9 6 ;.+-(, • P42 or less avorable events9 6 ;.+;7.



Chapter 2

27

SAN CAR$OS MU" S$I"ES Consider the decision tree on pa"e 2:. Bccordin" to this tree, the optimal strate"& is to use the test as lon" as its cost is less than 2,217 6 EM4Test9 < EM4=ot test9 6 <7,7( < 1;,;;;. I the result o the test is positive, then it is better to build the retainin" !all. I the result o the test is ne"ative, then it is better not to build the !all. I the test is not ta*en, then do not build the !all.



Chapter 2

2

'et = denote the event that the result o the test is ne"ative and  denote the event that the slide occurs. =otice that P4not = G 9 6;.:, P4= G not 9 6 ;.-, and P49 6 ;.:. To compute the probabilities on the branches !e do the ollo!in"# • P4=9 6 P4= G 9P49 8 P4= G not 9P4not 9 6 4;.194;.;19 8 4;.-94;.::9 6 ;.+2• P4 G =9 6 4;.194;.;194;.+2-9 6 ;.;;12 • P4 G not =9 6 4;.:94;.;194;.1-7-9 6 ;.;/



Chapter 2

2:



Chapter 2

(;

(RA%#IC COR%ORATION Consider the decision tree belo!. Bccordin" to this tree, the optimal decision strate"& is to install the Q<'B= usin" normal production techniDues, !ith an EM o /(7,;;;. 'et $, , and T denote the event !e "et a positive, neutral, and ne"ative result rom the test, respectivel&. 'et Q denote the event that the Q<'B= installation is successul, and 0 that it is not successul. To compute the probabilities on the branches o the tree, !e do the ollo!in"# • P4$9 6 P4$ G Q9P4Q9 8 P4$ G 09P409 6 4;.794;.:29 8 4;.1-94;.;9 6 ;./-/, • P49 6 P4 G Q9P4Q9 8 P4 G 09P409 6 4;.294;.:29 8 4;.194;.;9 6 ;.1:2, • P4T9 6 P4T G Q9P4Q9 8 P4T G 09P409 6 4;.194;.:29 8 4;.7-94;.;9 6 ;.1-2, • P40 G $9 6 4;.1-94;.;94;./-/9 6 ;.;1(, • P40 G 9 6 4;.1;94;.;94;.1:29 6 ;.;+17, • P40 G T9 6 4;.7-94;.;94;.1-29 6 ;.(:+7.


DMDch02

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