SM230 – Spring 1997 Practice Final, Solutions

4/29/97 – J. Turner

#1

 

Reg

Mid

Prem

Total

Fill

.125

.09

.28

.495

Not Fill

.375

.06

.07

.505

Total

.5

.15

.35

1

Cond.

.25

.6

.8

 
  1. See above
  2. 0.495
  3. 0.28
  4. 0.25
  5. Negative, P(Reg|Fill)<P(reg)

#2

Company A

 

AP

HE

AA

 

Defect

0.016

0.0025

0.0035

0.022

Not

 

 

 

 

Total

0.4

0.25

0.35

 

Cond

0.04

0.01

0.01

 

 

Company B

 

AP

HE

AA

 

Defect

0.008

0.005

0.0105

0.0235

Not

 

 

 

 

Total

0.40

0.25

0.35

 

Cond

0.02

0.02

0.03

 

 

#3

 

Reg

Res

Draftee

Total

Surrender

0.015

0.060

0.090

0.165

Not

0.2858

0.340

0.210

0.835

Total

0.30

0.40

0.30

 

Cond

0.05

0.15

0.30

 

 

  1. 16.5%
  2. P(Draft|surr)=0.09/0.165=0.545
  3. P(Surr|Draft) < P(Surr), so not independent

#4 X=# hydrophones that hold out of 6

1-BCDF(p=0.65,N=6,X=4) = 0.3191

#5

  1. BCDF(p=0.15,N=150,X=19) = 0.2509
  2. Solve BCDF(0.15,N,X=19) = 0.01. N=206 -> Prob = 0.00976

#6

  1. X=# Type O. Prob(X=15)=1-Prob(X<=14) = 1-HCDF(G=30,B=20,S=15,X=14) = 0.000069
  2. P(X>=8)=1-P(X<=7) = 1-HCDF(G=30,B=20,S=15,X=7) = 0.8278
  3. Prob(6<=X<=10)=HCDF(X=10)-HCDF(X=5) = 0.8131

#7

  1. PCDF(L=1,T=24,X=17)=0.087; so Prob(18 or more)=0.913
  2. Prob(X<7|X>5)=Prob(X<2)=1-PCDF(L=1,T=2,X=0) = 0.8647

#8

  1. PCDF(L=1/10,T,X=8)=0.90 -> T=54.25 miles
  2. PCDF(L,100,8)=0.90 -> L=0.05425/mile

#9

x

p(x)

y

p(y)

0

0.09

0

0.42

1

0.33

1

0.46

2

0.41

2

0.12

3

0.17

 

 

 

w

p(w)

0

0.0385

1

0.18263

2

0.335728

3

0.297034

4

0.125727

5

0.020381

 

  1. 0<=W<=5
  2. P(W<4)=P(W<=3) = 0.0385+0.18263+0.335728+0.297034=0.8539
  3. P(W>=2)=1-P(W<=1)=1-(0.0378+0.18)=0.7789
  4. P(W<=3)=0.8539
  5. mean=1.65
  6. mean(w)=2.56

#10 4 Frigate engines, p=0.20, 4 cruiser engines, p=0.15

  1. Total =0 -> Prob(F=0)*Prob(C=0)=0.4096*0.522=0.2138
  2. Total = 2 -> (F=0,C=2)+(F=1,C=1)+(F=2,C=0) = 0.27106

#11

  1. Total=0 -> 0.5*0.2=0.1
  2. Prob(T>=3)=P(R=3)+P(T=4) = 0.18+0.04=0.22
  3. Mean(X)=0.7, Mean(Y)=01.0, Mean(T)=0.7+1.0=1.7, Mean (20 lines)=20*1.7=34

SD(X)=sqrt(1.1-0.7^2)=0.781, SD(Y)=sqrt(1.4-1^2)=0.6325, SD(T)=1.005, SD(20)=4.494

  1. For sum of 30, mean=30*1.7=51, SD=5.5046, M+/-2S = 39.99 -> 62.01

#12

  1. Mean hits by 1 gun = 0.3, by 40 guns = 12.
  2. StdDev(1 gun) = 0.6708, StdDev(40 guns) = 4.2426
  3. 1-ZCDF(X=13.5) = 1-0.63816
  4. 1-ZCDF(20.5)=1-0.9774
  5. ZCDF(17.5)=0.9026, so 17 or fewer hits
  6. 0.63816^3=0.2599

#13

  1. 5
  2. sqrt(13)=3.606
  3. sqrt(6^2+8^2)=sqrt(100)=10