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PBT501S - PROBABILITY THEORY 1 - 1ST OPP - NOVEMBER 2023 |
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nAmlBIA UOIVERSITY
OF SCIEnCEAnDTECHnOLOGY
FacultyofHealth,Natural
ResourcaensdApplied
Sciences
Schoolof NaturalandApplied
Sciences
Departmenot f Mathematics,
StatisticsandActuarialScience
13JacksonKaujeuaStreet T: +264612072913
Private Bag13388
E: msas@nust.na
Windhoek
W: www.nust.na
NAMIBIA
QUALIFICATION : BACHELOR of SCIENCE
QUALIFICATIONCODE: 07BOSC
COURSE:PROBABILITY THEORY 1
LEVEL:5
COURSECODE: PBTS0lS
DATE: NOVEMBER 2023
SESSION: 1
DURATION: 3 HOURS
MARKS: 100
EXAMINER:
MODERATOR:
FIRST OPPORTUNITY: QUESTION PAPER
Dr, Dismas Ntirampeba
Mr, Jonas Amunyela
INSTRUCTIONS;
1. Answer all questions on the separate answer sheet.
2. Please write neatly and legibly.
3. Do not use the left side margin of the exam paper. This must be allowed for the
examiner.
4. No books, notes and other additional aids are allowed.
5. Mark all answers clearly with their respective question numbers.
PERMISSIBLE MATERIALS;
1. Non-Programmable Calculator
ATTACHEMENTS
1. Z Table
This paper consists of 4 pages including this front page
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Question 1 [20 marks]
1.1 Briefly explain the following terminologies as they are applied in set theory and prol.>ability
theory.
(i) Set
[2]
(ii) Random experiment
[2]
(iii) Sample space
[2]
(iv) Event
[2]
(v) Independent events (say A and B)
[2]
1.2 Consider the events A = {l, 3}, B = {2, 5}, C = {6}, D = {3, 4} over a possibility space
S = {l, 2, 3, 4, 5, 6}.With reasons, state whether true or false
(i) Events A,B and C are pairwise mutually
[2]
(ii) {A, B, C, D} is a partition of the sample space S
[3]
(iii) 0c= {1, 2, 3, 4, 5}
[2]
(iv) .4.6..C= {1,3}
[3]
Question 2 [20 marks)
2.1 Suppose that there were 120 students in the classroom, and that they could be classified as
follows:
Gender
Male
Female
Total
Hair color
Brown
20
-
50
Not Brown
-
30
-
Total
60
-
120
Let A= "student has brown hair" and B="student is male". Find:
(i) P(A n B)
[2]
(ii) P(A u B)
[2]
(iii) P(A I B)
[3]
2.2 In a certain federal prison, it is known that 2/3 of the inmates are under 25 years of age. It
is also known that 3/5 of the inmates are male and that 5/8 of the inmates are female or 25
years of age or older. What is the probability that a prisoner selected at random from this
prison is female and at least 25 years old?
[5].
2.3 A certain federal agency employs three consulting firms (A, B, and C) with probabilities
0.40, 0.35, and 0.25, respectively. From past experience it is known that the probability of
cost overruns for the firms are 0.05, 0.03, and 0.15, respectively. Suppose a cost overrun is
experienced by the agency.
(i) What is the probability that the consulting firm involved is company C?
[5]
(ii) ·what is the probability that the consulting firms involved are both A and C
companies?
[3]
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4.3 Mensa is an organisation whose members possess IQs that are in top 2% of the population. It
is known that the IQs arc normally distributed with a mean of 100 and a standard deviation
of 16.
(i) Find the proportion of Mensa members who has an IQ greater than 120.
[3]
(ii) Find the minimum IQ needed to be a Mensa member.
[4]
4.4 The jurisdiction of a rescue team includes emergencies occurring on a stretch of river that
is 4 miles long. Experience has shown that the distance along this stretch, measured in
miles from its northernmost point, at which an emergency occurs can be represented by a
uniformly distributed random variable over the range O to 4 miles. Then, if X denotes the
distance (in miles) of an emergency from the northernmost point of this stretch of river,
(i) find E(X),
[2]
(ii) find P(X 2.5).
[4]
4.5 A professor sees students during regular office hours. Time spent with students follows an
exponential distribution with a mean of 10 minutes.
(i) Find the probability that a given student spends fewer than 20 minutes
with the professor.
[4]
(ii) Find the variance of the time spent with students.
[2]
END OF QUESTION PAPER
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Standard Normal Probabilities
z Table entry for is the area under the standard normal curve
to the left of z.
z
.00
. 01
.02
-3.4 .0003 .0003
-3.3 .0095 .Q005
-3.2 .0007 .0007
-3.1- .0010 -:-0009
-3.0 .0013 .0013
-2.9 .001..2_ .6618
-2.8 .0026 .0025
-2.7 .0035 .0034
-2.6 .0047 .0045
-2.5 .0062 .0060
-2.4 .0082 .0080
-2.3 .0107 .0104
-2.2 .0139 .0136
-2.1 .0179 .0174
-2.0 .0228 .0222
-1.9 .0287- .0281
-1.8 .0359 .0351
-1.7 .0446 .0436
-1.6 .0548 .0537
-1.5- .0_668 .06~5
-1.4 .0808 .0793
_-13 .0968 .0~5L
- -1.2 .1151 .1131
-1.1 .1357 .1335
-1.0 .1587 .1562
-0.9 .1841 -:-18!3
-0.8 .2119 .2090
-0.7 .2420 .2389
-0.6 .2743 .2709
=-o.5 .3085 .3050
-0.4
-0.3
.3446
.3821
..33470893-
-0.2 .4207 .4168
-0.~- .4602 .4562
-0.0 .5000 .4960
.0003
.0005
.0006
.0009
.0013
.0018
.0024
.0033
.0044
.0059
.0078
.0102
.0132
.0170
.0217
.0274
.0344
.0427
.0526
.0643
.0778
.0934
.1112
.1314
.1539
.1788
.2061
.2358
.2676
.3015
.3372
.3745
.4129
-1522
.4920
.03
.04
.0003 .0003
.0004 .0004
.0006 .0006
.OOQ~ .0008
.0012 .0012
.0017 .0016
.0023 .0023
.0032 .0031
.0043 .0041
.0057 .0055
.0075 .0073
.0099 .0096
.0129 .0125
.0166 .0162
.0212 .0207
.0268 .0262
.0336 .0329
.0418 .0409
.0516 .0505
.0630 .0618
.0764 .0749
.0918 .0901
.1093 .1075
.1292 .1271
.1515 .1492
.1762 .1736
.2033 .2005
.2327 .2296
.2643 .2611
.2981 .2946
.3336 .3300
.3'Z_07 .3669
.4090 .4052
.4483 .4443
.4880 .4840
.OS
.0003
.0004
.0006
.0008
.0011
.0016
.0022
.0030
.0040
.0054
.0071
.0094
.0122
.0158
.0202
.0256-
.0322
.0401
.0495
.0606
.0735
.0885
.1056
.1251
.1469
.1711
.1977
.2266
.2578
.2912
.3264
.3632
.4013
.4404
.4801
.06
.0003
.0004
.0006
.0008
.0011
.QQ15
.0021
.0029
.0039
.0052
.0069
.0091
.0119
.0154
.0197
.0250
.0314
.0392
.0485
.0594
.0721
.0869
.1038
.1230
.1446
.1685
.1949
.2236
.2546
.2877
.3228
.3594
.3974
.4364
.4761
.07
.0003
.0004
.0005
.0008
.0011
.0015
.0021
.0028
.0038
.0051
.0068
.0089
.0116
.0150
.0192
.0244
.0307
.0384
.0475
.0582
.0708
.0853
.1020
.1210
.1423
.1660
.1922
.2206
.2514
.2843
.3192
.3557
.3936
.4325
.4721
.08
.0003
.0004
.0005
.0007
.0010
.0014
.0020
.0027
.0037
.0049
.0066
.0087
.0113
.0146
.0188
.0239
.0301
:0375
.0465
.0571
.0694
.0838
.1003
.1190
.1401
.1635
.1894
.2177
.2483
.2810
.3156
.3520
.3897
.4286
.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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Standard Normal Probabilities
z Table entry for is the area under the standard normal curve
z
to the left of z.
z .oo .01
.02
.03
.04
.OS
.06
.07
.08
.09
0.0 .5000 .5040 .5080
0.1 .5398 .5438 .5478
0.2 .5793 .5832 .5871
0.3· .6179 .6217 .6255
0.4 .6554 .6591 .6628
0.5 .6~1f. .6950 .69_8~-
0.6 .7257 .7291 .7324
0.7 .7580 .7611 .7642
0.8 .7881 .7910 .7939
0.9 .8159 .. 8186 .8212
1.0 .8413 .8438 .8461
1.1 .8643 :8665 .8686
1.2 .8849 .8869 .8888
1.3 .9032 -~9049 .9066
1.4 .9192 .9207 .9222
1.5 .9332 .9345 .9357
1.6 .9452 .9463 .9474
1.7 .9554 .9564 .9573
1.8 .9641 .9649 .9656
1.9 .9713 .9719 .9726
2.0 .9772 .9778 .9783
2.,;:_ .9821 .9826 .9830
2.2 .9861 .9864 .9868
.2.3 ~.~893 .9896 .9898
2.4 .9918 .9920 .9922
2.5 .993~ .9940 -.9941
2.6 .9953 .9955 .9956
2.7 .9.965 .9966 .9967
2.8 .9974 .9975 .9976
2.9 .9~81 .9982 .9982
3.0 .9987 .9987 .9987
3.1 .9990 .9991 .9991
3.2 .9993 .9993 .9994
33 .9995 _ .9995 ]995
3.4 .9997 .9997 .9997
.5120 .5160
.5517 .5557
.5910 .5948
.6293 .6331
.6664 .6700
.7019 .7054
.7357 .7389
.7673 .7704
.7967 .7995
.8238 .8264
.8485 .8508
.8708 .8729
.8907 .8925
.9082 .9099
.9236 .9251
.9370 .9382
.9484 .9495
.9582 .9591
.9664 .9671
,9?32 .9738
.9788 .9793
.9834 .9838
.9871 .9875
.9901 .9904
.9925 .9927
.9943 .9945
.9957 .9959
.9968 .9969
.9977 .9977
.9983 .9984
.9988 .9988
.999L__ .9992
.9994 .9994
.9996 .9996
.9997 .9997
.5199
.5596
.5987
.6368
.6736
.70_88
.7422
.7734
.8023
.8289
.8531
.8749
.8944
.9115
.9265
- .9394
.9505
.9599
.9678
.9744
.9798
.9842
.9878
.9906
.9929
.9946
.9960
.9970
.9978
.9984
.9989
.9912
.9994
.9996
.9997
.5239
.5636
.6026
.6406
.6772
.7123
.7454
.7764
.8051
.8315
.8554
.8770
.8962
.9131
.9279
.9406
.9515
.9608
.9686
.9750
.9803
.9846
.9881
.9909
.9931
.9948
.9961
.9971
.9979
.9985
.9989
.9992
.9994
.9996
.9997
.5279
.5675
.6064
.6443
.6808
.7157
.7486
.7794
.8078
.8340
.8577
.8790
.8980
.9147
.9292
.9418
.9525
.9616
.9693
.9756
.9808
.9850
.9884
.9911
.9932
.9949
.9962
.9972
.9979
.9985
.9989
.9992
.9995
.9996
.9997
.5319 .5359
.5714 .5753
.6103 .6141
.6480 .6517
.6844 .6879
.7190 .7224
.7517 .7549
.7823 .7852
.8106
.8365
-
.8133
.8389
.8599 .8621
.8810 .8830
.8997 .9015
.9162 .9177
.9306 .9319
.9429 .9441
.9535 .9545
.9625 .9633
.9699 .9706
.9?61 .9767
.9812 .9817
.9854 .9857
.9887 .9890
.9913 .9916
.9934 .9936
.9951 .9952
.9963 .9964
.9973 ~9974
.9980 .9981
.9986 .9986
.9990 .9990
.9993 .9993
.9995 .9995
.9996 .9997
.9997 .9998