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PBT501S - PROBABILITY THEORY 1 - 2ND OPP - JULY 2022 |
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NAMIBIA UNIVERSITY
OF SCIENCE AND TECHNOLOGY
FACULTY OF HEALTH, APPLIED SCIENCES, AND NATURAL
RESOURCES
DEPARTMENT OF MATHEMATICS AND STATISTICS
QUALIFICATION : BACHELOR OF SCIENCE APPLIED MATHEMATICS AND STATISTICS
QUALIFICATION CODE: 07BAMS
LEVEL: 5
COURSE: PROBABILITY THEORY 1
COURSE CODE: PBT501S
DATE: JULY 2022
SESSION: JULY
DURATION: 3 HOURS
MARKS: 100
SUPPLEMENTARY/ SECOND
EXAMINER(S)
Dr. D. Ntirampeba
Mr. E. Mwahi
OPPORTUNITY
EXAMINATION
QUESTION
PAPER
MODERATOR:
Mr. A. Roux
THIS QUESTION PAPER CONSISTS OF 4 PAGES
(Excluding this front page and statistical tables)
INSTRUCTIONS
1. Answer ALL the questions.
Write clearly and neatly.
3. Number the answers clearly.
PERMISSIBLE MATERIALS
1. Non-programable calculator
ATTACHMENTS
1. Statistical tables (Z-Tables)
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Question 1 [25 marks]
1.1
Assume that {B,, B,,...,B,}is a partition of S such that P(B,)> 0, for /=1,2,...,k.
k
Prove that for any event A in the sample S, P(A) = > P(A|B,) P(B,)
[5]
i=]
1.2
Let X be a binomial random with a probability mass function given by
f@=
n
(\\)p
k ,.n-x
_
q
,forx =0,1,..,n
0,
elsewhere
Show that E(X) = np.
[10]
1.3
Dr.Richmond, a psychologist, is studying the daytime television
students. She believes 45 percent of college students watch
afternoon. To further investigate, she selects a sample of 10.
viewing habits of college
soap operas during the
1.3.1 Write down a probability distribution for the number of students in the sample who
watch soap operas.
[3]
1.3.2 Find the mean and variance of this distribution.
[4]
1.3.3. What is the probability of finding exactly four watch soap operas?
[3]
Question 2[30 marks]
2.1
Consider the experiment of tossing a fair coin three times.
2.1.1 Develop a tree diagram for the experiment.
[2]
2.1.2 List the experimental outcomes.
[8]
2.1.3. What is the probability for each experimental outcome?
[1]
2.2. A committee of 5 persons is to be formed from 6 men and 4 women. In how many
ways can this be done when:
2.2.1 at least 2 women are included?
[4]
2.2.2 at most 2 women are included?
[4]
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2.3
The following information is based on the trends in the United States published by
the Food Marketing Institute, Washington, D.C. The columns represent length of
customer loyalty (in years) at a primary supermarket. The rows represent regions in
United States
Region
East
Midwest
South
West
Column
total
Less than a
yel-ar
32
31
53
41
1-2
years
54
68
92
56
Loyalty
3-4
years
59
68
93
67
5-9
years
112
120
158
78
10-14
years
77
63
106
45
15 or more
years
118
173
158
86
Row
total
452
523
660
373
157
270
287
468
291
535
2008
What is the probability that a customer chosen at random
2.3.1 has been loyal at least 10 years or is from Midwest?
[2]
2.3.2 has been loyal at least 10 years, given that he or she is from South or West?
[3]
2.4.
A large industrial firm uses 3 local motels to provide overnight accommodations for its clients.
From past experience it is known that 20% of clients are assigned rooms at the Ramada Inn,
50% at Sheraton, and 30% at Lakeview Motor Lodge. If the plumbing is faulty in 5% of the
rooms at Ramada Inn, in 4% of the rooms at Sheraton, and in 8% of the rooms at Lakeview
Motor Lodge, what is the probability that
2.4.1 aclient will be assigned a room with faulty plumbing?
[3]
2.4.2 a person with a room having faulty plumbing was assigned an accommodation at
Lakeview Motor Lodge?
[3]
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Question 3 [20 marks]
3.1. A The probability distribution of X, the number of imperfections per 10 meters of a
synthetic fabric in continuous rolls of uniform width, is given by
x
o| 1a] 2] 3] 4
p(x) | 0.41 | 0.37 | 0.16 | 0.05 | 0.01
3.1.1.
3.1.2.
What is the probability of no imperfections in 10 meters of a synthetic fabric? [1]
What is the probability that there are at least three imperfections in 10 meters of a
synthetic fabric?
[2]
3.1.3.
3.1.4.
3.1.5.
3.1.6.
What is the expected number of imperfections in 10 meters of a synthetic fabric?
[3]
What is the standard deviation of the imperfections in 10 meters of a synthetic
fabric?
[3]
Compute the coefficient of variation for the imperfections in 10 meters of a
synthetic fabric?
[2]
Construct the cumulative distribution of X and hence find the median of X
[4]
3.2 Arandom variableX has amean pw = 10 anda variance a? = 4. Use Chebyshev’s theorem
to estimate P(|X — 10| = 3)
[5]
Question 4 [25 marks]
4.1
LetX be random variable with a probability mass function given by
fi) = seHr u* fer x=0,1,2...,
0,
elsewhere
Show that )3% f(x) =1
[5]
4.2 The number of typing errors made by a typist has Poisson distribution with an average
of four errors per page. If more than four errors on a given page, the typist must retype
the whole page.
4.2.1 What is the probability that a certain page does not have to be retyped?
[5]
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4.2.2 What is the expected number of typing errors will the typist make in5 pages? [2]
4.2.3. What is the standard deviation number of typing errors will the typist make in 5 pages
[3]
4.3
University and college students average 7.2 hours of sleep per night, with a standard deviation
of 40 minutes. If the amount of sleep is normally distributed,
4.3.1 What proportion of university and college students sleep for more than 8 hours? [5]
4.3.2 Find the amount of sleep that is exceeded by only 25% of students.
[5]
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Standard Normal Probabilities
Table entry
Table entry for z is the area under the standard normal curve
:
to the left of z.
Zz
-00
-O1
.02
-3.4 .0003
.0003 .0003
=3:3,.0005
-..0005. _-.0005
-3.2 .0007
.0007 .0006
=3H. 0010 = .0009 = .0009
-3.0 .0013
=2:9 0019
0013 = .0013
0018 0018
2.8 .0026° .0025 .0024
-2.7 .0035
.0034 = .0033
-2.6
.0047
.0045 .0044
2.5
.0062 .0060 .0059
—2.4 .0082
.0080 .0078
=2.377.0107:--.
0104 =) 0102
—2.2
.0139
0136 =. .0132
—2:1 .0179
.0174 ~=.0170
-2.0 .0228
.0222 .0217
1.9) (0287. 0281 © 0274
-1.8 .0359
.0351
.0344
-1.7 .0446
.0436
8.0427
-1.6 .0548
.0537
#.0526
=l.5 ..0668. . .0655. 0643
-1.4 .0808
.0793
~.0778
ml.3 3.0968. 0951) 0934
—1.2
.1151
1131 = 1112
Slee 357.
1335. 1314
-1.0 .1587
1562
.1539
0.9 .1841
1814
=.1788
0.8 .2119
.2090 =. .2061
—0.7. .2420 3 §8.2389 =. .2358
0.6 .2743
.2709 ~—-.2676
=0:5° .3085
3050%
33015
0.4 .3446
.3409 = .3372
—0.3 .3821
3783
3745
0.2
.4207
4168 = .4129
SOs 4602.
4562
~=—.4522
—0.0 .5000
4960
.4920
-03
.04
.05
.06
.07
-08
.0003
.0003.
=.0003 .0003
.0003 #8 .0003
0004
.0004
.0004
.0004
.0004
.0004
.0006
.0006
§=«.0006 86.0006 .0005
#£.0005
0009
~=.0008 0008
.0008 .0008 .0007
.0012
0012
.0011 .0011
.0011
.0010
“= .0017 0016 0016 {0015 = 0015 = .0014
.0023
0023.
»=©.0022 )»3=.0021 .0021
.0020
.0032
§.0031
0030
.0029
.0028
.0027
.0043
.0041
.0040
.0039
.0038
#.0037
.0057
0055
.0054 .0052
0051
.0049
.0075
.0073
.0071
.0069
.0068
.0066
0099
.0096
.0094 .0091
.0089
.0087
.0129
0125.
.0122) =.0119
0116
.0113
0166
=©©.0162 §=.0158 .0154
0150
.0146
0212
.0207
.0202 .0197
.0192
.0188
0268
.0262
.0256
.0250
.0244
.0239
0336
=©.0329 =.0322) =.0314 ~=—.0307_~——.0301
0418
.0409
.0401 .0392
.0384
#.0375
0516
.0505
.0495 .0485
.0475
.0465
0630.
.0618
.0606 ~.0594
.0582° = .0571
.0764
.0749
.0735
.0721
.0708
.0694
0918
.0901
.0885
.0869
.0853 + .0838
-1093
.1075
1056
.1038
.1020
.1003
L292
27d
1251
230"
12101190
1515
1492
.1469 .1446
.1423
.1401
“1762?
1736.
A711
1685.
21660-1635
.2033
2005
.1977 .1949
.1922
.1894
123278 + 229072000.
.2250
2200"
2177
.2643
.2611
2578
=©.2546 38.2514
~=—.2483
2981
2946;
2.2912. 32877... 7.2843, = 2810
3336
=..3300)3Ss «.3264)=S ss 3228 )=3)—S 3192 ~—— 3156
33/0/. .2,36609> 3632.) 3594. 335572. 43520
4090
4052
4.4013 .3974 # .3936
38.3897
4483
4443
~=.4404 = 4364 ~—S (i 44325~—Ss«w 44286
.4880
.4840
.4801
.4761
.4721 ~~ .4681
.09
.0002
.0003
.0005
.0007
.0010
0014
.0019
0026
.0036
0048
.0064
.0084
.0110
0143
.0183
.0233
.0294
.0367
.0455
.0559
.0681
.0823
.0985
.1170
.1379
1611
.1867
.2148
2451
.2776
3121
3483
3859
4247
.4641
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Table entry
Standard Normal Probabilities
Table entry for z is the area under the standard normal curve
to the left of z.
Z
.00
01
02
.03
.04
.05
.06
.07
.08
.09
0.0 .5000 .5040 .5080 .5120 .5160 .5199 .5239 5279 5319 5359
One 5398 5438 | 25478 5517 5557, 5506 5636. 5675 5714 5753
0.2 .5793 .5832 5871 .5910 .5948 .5987 .6026 .6064 .6103 .6141
0.33 6179
6217 © 6255 = .6293
6331 = «6368 =.6406 = 66443) =. 6480
6517
0.4 .6554 .6591 .6628
.6664 6700 .6736 4.6772 + «2.6808
.6844 ~ .6879
05.2 6915 = 6050 6985. = 7019 7054. = 7088 =~ £7123 7157 7190. 7224
0.6 .7257
.7291 .7324 .7357 =.7389. = .7422,)—Ss:«w7454—S 74867517
-~——«.7549
(0:7) 7580 ell 7042 7673. 2 7704) 7734" 7764 7794. 7828. 782.
0.8 .7881 .7910 .7939 .7967 .7995 .8023
.8051 .8078 .8106
.8133
0.9 .8159 .8186 4.8212 .8238 .8264 8289 .8315 .8340 .8365 .8389
1.0 .8413 .8438 .8461 .8485 .8508 .8531 .8554 4.8577 8599
=—«.8621
li. (8643 © (8665 = 8686: =.8708 § .8729 8749 18770 8790-8810" 8830
1.2 .8849 .8869 .8888
.8907
.8925 .8944 .8962 .8980 .8997 .9015
1322 (9032 0049-9066 = 9082. = .90990° 9115 0131 9147-9162 = 9177
1.4 9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9319
1.5 .9332 19345 .9357 (9370 .9382 9394 9406 9418 9429 .9441
1.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545
1.7. .9554 9564 = 9573. 9582. 9591 = 9599 9608 = «9616 = 69625 9633
1.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9706
197 9713<..9719.. 49726 = 9732. 9738 9744 = 9750. 9756 = 9761 = = 9767
2.0 .9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .9817
21
.9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .9857
2.2 .9861 .9864 .9868
.9871 .9875 .9878 .9881 .9884
.9887
.9890
2.3
.9893 .9896 .9898 9901 .9904 9906 .9909 9911 9913 9916
2.4 .9918 .9920 .9922 .9925 .9927 9929 .9931 .9932 .9934 .9936
2.5 .9938 .9940 .9941 9943 9945 9946 19948 9949 9951 9952
2.6 .9953 .9955 .9956 .9957 .9959 .9960 .9961 .9962 .9963 .9964
2.2. .9965.- 9966. .9967 = .9968 ~.9969 9970 = (9971 = .9972° 19973" 9974
2.8 .9974 .9975 .9976 .9977 .9977 .9978 .9979 .9979 9980 .9981
2.9 .9981 .9982 .9982 .9983 .9984 .9984 .9985 .9985 .9986 9986
3.0 .9987 .9987 .9987 .9988 .9988 .9989 .9989 .9989 .9990 9990
3.1 .9990 .9991 .9991 .9991 9992 .9992 9992 .9992 9993 9993
3.2 .9993 .9993 .9994 .9994 .9994 .9994 .9994 .9995 9995 9995
3.3. .9995 .9995 .9995 .9996 .9996 .9996 .9996 .9996 .9996
.9997
3.4 .9997 .9997 .9997 .9997 .9997 .9997 9997 .9997 .9997 .9998