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Cha pt e r4 I nt r oduc t i ont oPr o ba bi l i t y

Le a r ni ngObj e c t i ve s 1 .

Obt a i na na p pr e c i a t i ono ft her ol epr oba bi l i t yi n f or ma t i onp l a y si nt hed e c i s i o nma ki ngp r o c e s s .

2 .

Un de r s t a n dpr oba bi l i t ya sanume r i c a lme a s ur eoft hel i k e l i hoo dofo c c u r r e nc e .

3 .

Kno wt het hr e eme t hod sc ommon l yu s e df ora s s i gni ngpr oba bi l i t i e sa ndu nde r s t a ndwh e nt h e ys h oul d b eus e d.

4 .

Kno whowt ou s et hel a wst ha ta r ea v a i l a bl ef o rc o mpu t i n gt h epr oba bi l i t i e sofe v e nt s .

5 .

Un de r s t a n dho wne wi n f o r ma t i o nc a nb eus e dt or e vi s ei n i t i a l( p r i o r )pr ob a bi l i t ye s t i ma t e sus i n g Ba ye s ’t he or e m.

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Cha pt e r4

Sol ut i o ns : 1 .

Numbe rofe x pe r i me nt a lOut c ome s=( 3) ( 2 ) ( 4)=2 4

2 .

 6  6! 6 5 4 3 2 1  20    3  3!3! (3 2 1)(3 2 1) ABC ABD ABE ABF ACD

P36  3 .

ACE ACF ADE ADF AEF

BCD BCE BCF BDE BDF

BEF CDE CDF CEF DEF

6! ( 6)( 5)( 4) 120 ( 6  3)!

BDF BFD DBF DFB FBD FDB 4 . a . 1 s tT os s

2 ndT os s

3r dTo s s H

H T H

( H, H, H)

T ( H, H, T) H

( H, T, H)

T ( H, T, T) H

T H T

( T, H, H)

T ( T, H, T) H

( T, T, H)

T ( T, T, T) b. Le t :Hbeh e a da ndTbet a i l ( H, H, H)( T, H, H) ( H, H, T)( T, H, T) ( H, T, H)( T, T, H) ( H, T, T)( T, T, T) c . Theou t c ome sa r ee qu a l l yl i k e l y ,s ot hep r ob a bi l i t yo fe a c hou t c omei s1 / 8 . 5 .

P( Ei)=1 / 5f ori=1 ,2 ,3, 4,5 P( Ei) 0f o ri=1 ,2, 3, 4,5 P( E1)+P( E2)+P( E3)+P( E4)+P( E5)=1/ 5+1/ 5+1/ 5+1 / 5+1/ 5=1 4-2 © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

I n t r odu c t i ont oPr ob a bi l i t y

Th ec l a s s i c a lme t ho dwa sus e d . P( E1)=. 4 0,P( E2)=. 26,P( E3)=. 3 4

6 .

Th er e l a t i vef r e qu e n c yme t h odwa sus e d. 7 .

No .Re qu i r e me nt( 4 . 4 )i snots a t i s fie d ;t hepr ob a bi l i t i e sdonot s umt o1 .P1( ) E +P( E2)+P( E3)+ P( E4)=. 1 0+. 15+. 4 0+. 2 0=. 85

8 . a . The r ea r ef o urou t c ome spo s s i bl ef o rt hi s2s t e pe xp e r i me nt ;pl a n ni n gc ommi s s i o npos i t i v e-c o unc i l a ppr ov e s ;pl a n ni n gc ommi s s i onp os i t i ve-c o un c i ld i s a pp r o v e s ;p l a nni ngc o mmi s s i onn e g a t i v ec oun c i la pp r o v e s ;pl a nni n gc ommi s s i o nne g a t i v e-c oun c i ld i s a pp r o v e s . b. Le tp=pos i t i v e ,n=n e g a t i v e ,a=a pp r ov e s , a n dd=di s a pp r o v e s

P l a n n i n gCo mmi s s i o n

Co u n c i l a

( p ,a )

d p

( p ,d )

n a

.

( n ,a )

d ( n ,d )

9 .

 50  50! 50 49 48 47  230,300   4 4!46! 4 3 2 1  

10. a.

Using the table provided, 94% of students graduating from Morehouse College have debt. P(Debt) = .94

b. Five of the 8 institutions have over 60% of their graduates with debt. P(over 60%) = 5/8 = .625 c. Two of the 8 institutions have graduates with debt who have an average debt more than $30,000. 4-3 © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Cha pt e r4

P(more than $30,000) = 2/8 = .25 d.

P(No debt) = 1 - P(Debt) = 1 - .72 = .28

e. This is a weighted average calculation. 72% graduate with an average debt of $32,980 and 28% graduate with a debt of $0. .72($32, 980) .28($0) = $23,746 .72  .28 Average debt per graduate =

1 1.a . To t a lmot or c y c l i s t s=35 0+17 0=52 0 350 .6731 P( DOTCo mpl i a n tHe l me t )=520 b . Ye s ,t h eo v e r a l lpr oba b i l i t yh a sbe e ni n c r e a s i n gf r o m. 4 8fiv ey e a r sa g o ,t o. 63on ey e a ra g o , a ndi s n o wa ppr o x i ma t e l y. 67.Thepr o ba bi l i t yt h a tamot o r c y c l i s twe a r saDOTc o mpl i a nthe l me ta pp e a r s t ob ei nc r e a s i n g .

96 .6076 158 c . No r t he a s t : 86 .6667 Mi d we s t : 129

So ut h :

92 .6525 141

We s t :

76 .8261 92

Th eWe s tr e g i o ns h o wst hehi gh e s tpr oba bi l i t y( . 8 261 )ofDOTc omp l i a nth e l me tus e . 1 2.a . Us et hec o un t i n gr u l ef orc omb i n a t i on s :

 55  55! (55)(54)(53)(52)(51)  3,478, 761   (5)(4)(3)(2)(1)  5  5!50! Onec h a nc ei n3, 48 9, 761 b . Ve r ys ma l l :1/ 3, 47 8, 76 1=. 0 00 000 287 c . Mul t i pl yt hea ns we ri npa r t( a )b y4 2t og e tt hen umb e rofc ho i c e sf ort h es i xnu mbe r s . Nu mb e ro fCh oi c e s=( 3, 478 , 7 61 ) ( 4 2)=1 46, 10 7, 96 2 Pr oba bi l i t yofWi nni n g=1/ 1 46, 1 07, 96 2=. 000 00 000 684

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I n t r odu c t i ont oPr ob a bi l i t y

1 3.

I ni t i a l l yap r o ba b i l i t yo f. 2 0wo ul dbea s s i g ne di fs e l e c t i o ni se qu a l l yl i k e l y .Da t adoe sno ta p pe a rt o c onfir mt h ebe l i e fofe qu a lc on s ume rp r e f e r e nc e .Fo re x a mp l eus i n gt h er e l a t i v ef r e q u e nc yme t ho d wewo ul da s s i g napr oba bi l i t yof5 / 1 00=. 0 5t ot hede s i g n1o ut c o me , . 1 5t ode s i g n2, . 3 0t o d e s i gn3,. 40t od e s i g n4,a nd. 10t od e s i gn5.

1 4.a . P( E2)=1/ 4 b . P( a n y2o ut c ome s )=1/ 4+1/ 4=1 / 2 c . P( a n y3ou t c ome s )=1 / 4+1/ 4+1 / 4=3/ 4 1 5.a . S={ a c eofc l u bs , a c eofd i a mon ds , a c eo fh e a r t s , a c eo fs p a de s } b . S={2o fc l ub s ,3o fc l ub s ,... ,10o fc l ub s , Jofc l ub s , Qofc l ub s , Kofc l ub s , Aofc l ub s } c . The r ea r e12;j a c k,q u e e n,o rki n gi ne a c hoft hef o urs ui t s . d . Fo ra :4/ 52=1/ 13=. 0 8 Fo rb:13/ 5 2=1/ 4=. 25 Fo rc :12/ 5 2=. 23 1 6.a .( 6 ) ( 6)=36s a mpl ep oi nt s b . Di e2 1

2

3

4

5

6

1

2

3

4

5

6

7

2

3

4

5

6

7

8

3

4

5

6

7

8

9

4

5

6

7

8

9

1 0

5

6

7

8

9

1 0

1 1

6

7

8

9

1 0

1 1

1 2

T o t a lf o rBo t h

Di e1

c .6 / 36=1/ 6 d .1 0 / 3 6=5 / 1 8 4-5 © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

.

Cha pt e r4

e . No.P( o dd )=1 8/ 36=P( e v e n)=18 / 3 6or1 / 2f o rbo t h. f . Cl a s s i c a l . Apr o ba b i l i t yof1 / 36i sa s s i gn e dt oe a c he xp e r i me nt a lo ut c o me .

1 7.a .( 4 , 6 ) , ( 4, 7) ,( 4, 8) b .. 0 5+. 10+. 1 5=. 30 c .( 2 , 8 ) ,( 3, 8 ) ,( 4, 8) d .. 0 5+. 05+. 1 5=. 25 e .. 15 11 1 8.a . P( n ome a l s )=496 =. 02 22

b . P( a tl e a s t f ou rme a l s )=P( 4)+P( 5)+P( 6)+P( 7o rmo r e ) 36 119 114 139    496 496 496 496 =. 82 26 =

c . P( t woo rf e we rme a l s ) =P( 2 )+P( 1)+P( 0) 30

=496



11 496



11 496 =. 104 8

1 9.a . A summary of the data provided in the exercise follows: Response Yes No Unsure Total

United States 187 334 256 777

Great Britain 197 411 213 821

Total 384 745 469 1598

Probability = 334/777 = .4299 b.

Probability = (411 + 213)/821 = .76

c.

Probability = (334 + 411) /1598 = .4662

d.

The probability that an investor in the United States thinks the government is adequately protecting investors is 187/777 =.2407; for investors in Great Britain the probability is 197/821 = .24. The two probabilities are almost identical; thus, there does not appear to be a difference between the perceptions of investors in these two countries with regard to the “Yes” response. However, in part (a) we showed that the probability that an investor in the United States does not think the government is adequately protecting investors is.4299, or approximately .43; for investors 4-6 © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

I n t r odu c t i ont oPr ob a bi l i t y

in Great Britain the probability is 411/821 = .5006 or approximately .50. These results show a slightly higher probability that an investor in Great Britain will say that the government is not protecting investors adequately. 2 0.a . P( N)=5 4/ 50 0=. 1 08 b . P( T)=4 8 / 50 0=. 096

c . To t a li n5s t a t e s=5 4+52+48+3 3+3 0=21 7 P( B)=21 7/ 50 0=. 4 34 Al mos th a l ft heFo r t une5 00c o mpa ni e sa r eh e a d qua r t e r e di nt he s efiv es t a t e s . P  Fall  

21. a.

6455 4535

 0.1422

P  Transportation Incident 

b. c.

1795 4535

 0.3958

The Cause of Fatality that is least likely to occur is Fires and Explosions with a probability of P  Fires and Explosions 

113 4535

 0.0249

2 2.a . P( A)=. 40, P( B)=. 4 0, P( C)=. 6 0 E4)=. b . P( A B)=P( E1,E2,E 80.Ye sP( A B)=P( A)+P( B) . 3, c . Ac={ E3,E E5}Cc={ E1,E4}P( Ac)=. 60P( Cc)=. 40 4, E1, E2,E5}P( 60 d . A Bc ={ A Bc)=. E3,E4,E5)=. 8 0 e . P( B C)=P( E2, 2 3.a . P( A)=P( E1)+P( E4)+P( E6)=. 05+. 2 5+. 10=. 40 P( B)=P( E2)+P( E4)+P( E7)=. 2 0+. 2 5+. 05=. 5 0 P( C)=P( E2)+P( E3)+P( E5)+P( E7)=. 2 0+. 2 0+. 15+. 0 5=. 60 E6,E7} b . A B={ E1,E2,E 4, E2)+P( E4)+P( E6)+P( E7) P( A B)=P( E1)+P( =. 05+. 2 0+. 25+. 10+. 05=. 65 25 c . A B={ E4} P( A B)=P( E4)=. d . Ye s , t h e ya r emut ua l l ye x c l us i v e . 4-7 © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Cha pt e r4

e . Bc={ E1,E E5,E6} ;P( Bc) =P( E1)+P( E +P( E5)+P( E6) 3, 3) =. 0 5+. 2 0+. 15+. 1 0=. 50 2 4.

Le tE=e xpe r i e n c ee x c e e de de xp e c t a t i ons M =e xp e r i e nc eme te xp e c t a t i o ns

a . Pe r c e nt a g eo fr e s po nde nt st ha ts a i dt he i re xp e r i e nc ee x c e e de de x pe c t a t i on s =10 0-( 4+2 6+6 5)=5% P( E)=. 05 b . P( M  E)=P( M)+P( E)=. 65+. 05=. 7 0 2 5.

Le tM =ma l ey ou n ga dul tl i v i n gi nhi spa r e n t s ’home F =f e ma l ey o un ga du l tl i vi n gi nh e rp a r e nt s ’ho me

a . P( M  F)=P( M)+P( F)-P( M  F) =. 56+. 4 2-. 2 4=. 74 b . 1-P( M  F)=1-. 7 4=. 26 2 6.a . Le tD=Do me s t i cEq ui t yFund P( D)= 16 / 2 5=. 64 b . Le tA=4 -or5 s t a rr a t i n g 13 funds were rated 3-star of less; thus, 25 – 13 = 12 funds must be 4-star or 5-star. P( A)=1 2/ 25=. 4 8 c . 7Dome s t i cEq u i t yf u ndswe r er a t e d4 s t a ra n d2we r er a t e d5s t a r .Thus ,9f un dswe r eDo me s t i c Eq ui t yf und sa n dwe r er a t e d4 s t a ror5s t a r P( D A)=9 / 2 5=. 36 d . P( D A) =P( D)+P( A)-P( D A) =. 64+. 4 8-. 36=. 7 6 2 7.

Le t A=t hee v e n tt heACCha sat e a mi nt h ec ha mpi o ns h i pg a me S=t h ee v e ntt h eSECha sat e a mi nt hec h a mpi on s hi pg a me P (A ) 

10  .50 20

P (S ) 

8  .40 20

a .

b .

P (A  S ) 

c .

1  .05 20

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I n t r odu c t i ont oPr ob a bi l i t y

Th e r ei sal o wp r o ba b i l i t yt h a tt e a msf r o mbo t ht heACCa ndSECwi l lbei nt h ec ha mpi on s hi pg a me . d . P (A  S )  P (A )  P (S )  P (A  S ) .50  .40  .05 .85 Th e r ei sah i g hpr oba bi l i t yt ha tat e a mf r o mt h eACCo rSECwi l lbei nt h ec ha mp i o ns h i pg a me . 1  P( A  S ) 1  .85 .15 e . P( Ne i t h e rc onf e r e nc e )=

In this case, teams will most likely come from the Big Ten (6), Big East (4), Pac-10 (4), or Big 12 (3). Numbers shown are the number of times teams from these conferences have played in the national championship game over the previous 20 years.

2 8.

Le t : B=r e nt e dac a rf orb us i n e s sr e a s o ns P=r e nt e dac a rf o rpe r s o na lr e a s ons

a . P( B P)=P( B)+P( P)-P( B P) =. 54+. 4 58-. 30=. 6 98 b . P( Ne i t he r )=1-. 69 8=. 30 2 1033 .3623 29. a . P( E)=2851 854 .2995 2851 P( R)= 964 .3381 2851 P( D)=

b . Ye s ;P( E D)=0 1033 .4349 c . Pr ob a b i l i t y= 2375

d . Le tFd e no t et hee v e ntt ha tas t u de n twhoa p pl i e sf ore a r l ya d mi s s i o ni sd e f e r r e da n dl a t e ra dmi t t e d d ur i n gt her e gu l a ra d mi s s i onpr oc e s s . Ev e nt sEa ndFa r emut u a l l ye x c l us i v ea n dt h ea d di t i onl a wa pp l i e s . P( E F)=P( E)+P( F) P( E)=. 3 623f r o mpa r t( a ) Oft he9 64e a r l ya p pl i c a n t swhowe r ede f e r r e d,wee xp e c t1 8%,o r. 18( 964)s t u de n t s ,t ob ea dmi t t e d d ur i n gt her e gu l a ra d mi s s i onpr oc e s s .Th us , f o rt het o t a lo f2 851e a r l ya dmi s s i ona pp l i c a nt s .18(964) .0609 P( F)= 2851

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Cha pt e r4

P( E F)=P( E)+P( F)=. 362 3+. 06 09=. 42 32 No t e :. 1 8( 9 64 )=1 73 . 5 2.Somes t ude n t sma yr ou ndt h i st o1 74s t u de n t s .I fr ou nd i n gi sdo ne , t he a ns we rbe c ome s. 4 23 3.Ei t h e ra ppr oa c hi sa c c e p t a b l e . P (A B) 

P (A  B) .40  .6667 P (B) .60

P (B A) 

P (A  B) .40  .80 P (A) .50

3 0.a .

b .

c . Nobe c a us eP( A| B) P( A)

3 1.a . P( A B)=0 P (A B)  b .

P (A  B) 0  0 P (B) .4

c . No.P( A| B) P( A) ; t h ee v e nt s , a l t ho u ghmut ua l l ye x c l us i v e ,a r en oti n de p e nde n t . d . Mu t ua l l ye x c l us i v ee v e nt sa r ede pe nde n t . 3 2.a . Ro wa ndc ol umns umsa r es ho wn.

U.S. Non U.S. Total

Car 87.4 228.5 315.9

Light Truck 193.1 148.0 341.1

Total 280.5 376.5 657.0

At o t a lo f65 7. 0t ho us a ndv e hi c l e swe r es ol d. Di v i di n ge a c he nt r yi nt h et a b l eb y65 7. 0p r o v i d e st h ef ol l o wi n gj oi n tp r ob a b i l i t yt a bl e . Car Light Truck U.S. .1330 .2939 Non U.S. .3478 .2253 Total .4808 .5192 b . Le t U=U.S. ma n uf a c t u r e r N=NonU. S. ma n uf a c t ur e r C=Ca r L= Li gh tTr uc k

Total .4269 .5731 1.0000

Ma r g i n a lpr o ba b i l i t i e s :P( U)=. 42 69 P( B)=. 573 1 Th e r ei sah i g he rpr o ba b i l i t yt ha tt hev e hi c l ewa sn otma n uf a c t ur e db yaU.S.a u t oma k e r .I nt e r ms o fma r k e ts h a r e , nonU. S. a ut oma k e r sl e a dwi t ha57 . 3 %s h a r eofv e hi c l es a l e s .

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I n t r odu c t i ont oPr ob a bi l i t y

Ma r g i n a lpr o ba b i l i t i e s :P( C)=. 48 08 P( L)=. 519 2 Th el i g htt r uc kc a t e g or ywhi c hi n c l u de spi c k up, mi ni v a ns ,SUVsa ndc r os s o v e rmode l sha sas l i g ht l y h i g he rp r oba bi l i t y .Bu tt h et ype sofv e hi c l e sa r ef a i r l ye v e ns p l i t . P (C U )  c .

P (C U ) .1330  .3115 P (U ) .4269

P (L U ) .2939 P (L U )   .6885 P (L ) .4269

I fav e hi c l ewa sma nu f a c t ur e db yoneo ft heU. S.a ut oma k e r s ,t he r ei sahi g he rp r o ba b i l i t yi twi l lbe i nt h el i g htt r uc kc a t e g or y . P (C N )  d .

P (C N ) .3478  .6069 P (N ) .5731

P (L  N ) .2253 P (L N )   .3931 P(L) .5731

I fav e hi c l ewa sn otma n uf a c t ur e db yoneo ft h eU. S.a u t oma ke r s ,t h e r ei sahi gh e rpr ob a bi l i t yi twi l l b eac a r .

e .

P (U  L ) .2939 P (U L )   .5661 P (L ) .5192 I fav e hi c l ewa sal i gh tt r uc k ,t he r ei sbe t t e rt ha na5 0 5 0c ha nc et h a ti twa sma nuf a c t u r e db yon eof t h eU. S.a ut oma k e r s .

f . Th e r ei sah i g he rpr oba b i l i t y ,a ndt hu sal a r g e rma r k e ts h a r ef orn onU. S.a u t oma k e r s .Ho we v e r ,t h e U. S. a ut oma k e r sa r el e a de r si ns a l e sf o rt hel i gh tt r uc kc a t e g or y . 33. a. Undergraduate Major Intended Enrollment Status

Business

Engineering

Other

Totals

Full-Time

0.270

0.151

0.192

0.613

Part-Time

0.115

0.123

0.149

0.387

0.385

0.274

0.341

1.000

Totals b.

P(Business) = 0.385, P(Engineering) = 0.274, and P(Other) = 0.341, so Business is the undergraduate major that produces the most potential MBA students. P  Engineering | Full - Time  =

c. P  Full - Time | Business  = d. e.

P  Engineering  Full - Time  0.151  0.246 P  Full - Time  0.613

P Full - Time  Business 0.270  0.701 P  Business  0.385

Let A denote the event that student intends to attend classes full time in pursuit of an MBA degree, and let B denote the event that the student was an undergraduate Business major. Are events A and B independent? Justify your answer. 4-1 1 © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Cha pt e r4

For independence, we must have that (a) of this problem, we have

P  A  P  B  P  A  B 

; from the joint probability table in part

P(A) = 0.613 P(B) = 0.385 So

P A P B  0.387  0.385 0.236

But

P A  B 0.270 Because

P  A  P  B  P  A  B 

, the events are not independent.

3 4.a . Le tO =fli g h ta r r i v e sont i me Oc =fli g h ta r r i v e sl a t e S =So u t h we s tfli g ht U =USAi r wa y sfli g ht J =J e t Bl uefli g ht Gi v e n :P( O| S)=. 83 4 P( S)=. 4 0

P( O| U)=. 7 51 P( O| J )=. 70 1 P( U)=. 35

P( J )=. 2 5

P (O  S) P (S) P( O| S)=

 P( O S)=P( O| S) P( S)=( . 8 34) ( . 4)=. 33 36 Si mi l a r l y P( O U)=P( O| U) P( U)=( . 7 51) ( . 35 )=. 26 29 P( O J )=P( O| J ) P( J )=( . 701 ) ( . 2 5)=. 1 75 3 J o i ntpr oba bi l i t yt a bl e