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PBT501S - PROBABILITY THEORY 1 - 1ST OPP - NOVEMBER 2024 |
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nAm I Bl A un IVE RS ITY
OF SCIEnCE AnD TECHnOLOGY
FacultoyfHealthN, atural
ResourceasndApplied
Sciences
Schoool f NaturalandApplied
Sciences
Departmentof Mathematics,
StatisticsandActuarialScience
13JacksonKaujeuaStreet
Private Bag13388
Windhoek
NAMIBIA
T: +264612072913
E: msas@nust.na
W: www.nust.na
QUALIFICATION : Bachelor of Science in Applied Mathematics and Statistics & Bachelor
of Science
QUALIFICATION CODE: 07BSAM & 07BOS
LEVEL:5
COURSE:PROBABILITYTHEORY 1
COURSECODE: PBTS0lS
DATE: NOVEMBER2024
SESSION: 1
DURATION: 3 HOURS
MARKS: 100
EXAMINER:
MODERATOR:
FIRSTOPPORTUNITYE: XAMINATIONQUESTIONPAPER
Dr D Ntirampeba
Mr J 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. Write all answers clearly with their respective question numbers.
PERMISSIBLME ATERIALS:
1. Non-Programmable Calculator
ATTACHEMENTS
1. Z Table
This paper consists of 3 pages excluding this front page
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Question 1 [20 marks]
1.1 Briefly explain the follmving terminologies as they are applied in set theory and proiJaiJility
theory.
(i) Set
[2]
(ii) Disjoint sets (say A and B)
[2]
(iii) Random experiment
[2]
(iv) Sample space
[2]
(v) Pairwise mutually exclusive events (say A1, A2 , ... , Ak)
[2]
1.2 Consider the sample space S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and the events A
B = {3, 4, 5, 6, 7}, C = {2, 4, 6}, D = {1, 3, 6}, E = {1, 2, 5, 9}.
(i) Are A, B, C, D , and E collectively exhaustive? Explain why.
Assuming that all elements are equiprobable, find:
(ii) P(0 - S)
(iii) P(A u D)
(iv) P(C6.D) (Hint: 6, means symmetric difference)
{1, 2, 5, 6},
[2]
[2]
[2]
[4]
Question 2 [30 marks]
2.1 (i) How many distinct permutations can be made from the letters of the word ROWS? [2]
(ii) How many of these permutations start with the letter S followed by letter R?
[2]
(iii) State three axioms/rules of probability
[3]
2.2 Factory workers are constantly encouraged to practice zero tolerance when it comes to ac-
cidents in factories. Accidents can occur because the working environment or conditions
themselves are unsafe. On the other hand, accidents can occur due to carelessness or so-
called human error. In addition, the worker's shift, 7:00 A.M - 3:00 P.M. (day shift),
3:00 P.M - 11:00 P.M. (evening shift), or 11:00 P.M - 7:00 A.M. (graveyard shift), may be a
factor. During the last year, 300 accidents have occurred. The percentages of the accidents
for the condition combinations are as follows:
Shift
Day
Evening
Graveyard
Unsafe conditions Human error
5%
32%
6%
25%
2%
30%
If an accident report is selected randomly from the 300 reports,
(i) what is the probability that the accident occurred on the graveyard shift?
[2]
(ii) what is the probability that the accident occurred on either the evening or the graveyard
shift?
[3]
(iii)If an accident occurred on the evening, what is the probability that it was due to human
error?
[4]
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2.3 A diagnostic test for a disease is said to be 90% accurate in that if a person has the disease,
the test will detect it with probability 0.9. Also, if a person does not have the disease, the test
will report that he or she does not have it with probability 0.9. Only 1% of the population
has the disease in question. If a person is chosen at random from the population and the
diagnostic indicates that she has it, what is the probability that she does, in fact, have the
disease?
[5]
2.4 Three airlines serve a small town in Ohio. Airline A has 50% of the all scheduled flights,
Airline B has 30%, and Airline Chas the remaining 20%. Their on-time rates are 80%, 65%,
and 40%, respectively.
(i) If a plane has just left, what is the probability that it left on time?
[3]
(ii) If a plane has just left on time, what is the probability that it was airline A or
Airline B
[6]
Question 3 [20 marks]
3.1 A company's marketing and accounting departments have determined that if the company
markets its newly developed product, the contribution of the product to the firm's profit
during the next 6 months will be described by the following:
Profit contribution
-$5000
$10000
$30000
Probability
0.2
0.3
0.5
(i) Find the company's expected profit
[3]
(ii) Find the median company's profit (hint: First, find the cumulative distribution of the
profit)
[5]
3.2 A manufacturing company has developed a machine for cleaning carpet that is fuel-efficient
because it delivers carpet cleaner so rapidly. Of interest is a random variable Y , the amount
in gallons per minute delivered. It is known that the density function is given by
1 if 7 < y < 8
f(y) = { 0 otherwise
Find:
(i)E(Y)
[3]
(ii) E(Y 2)
[3]
(iii) Var(Y)
[2]
(iv) median of X (hint: First, find F(x)).
[4]
Question 4 [30 marks]
4.1 A new surgical procedure is successful with a probability p = 0.8. Assume the operation is
performed ten times and the results are independent of one another. Find:
(i) the expected number of successful operations;
[3]
(ii) the probability that fewer than 2 are successful.
[4]
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4.2 For a certain type of copper wire, it is known that, on the average, 1.5 flaws occur per
millimeter. Assuming that the number of flaws is a Poisson random variaiJle,
(i) what is the probability that no flaws occur in a certain portion of wire of length 0.005
mcle~?
(ii) what is the mean number of flaws in a portion of length 0.005 meters?
[3]
4.3 Scores on a Biology examination are assumed to be normally distributed with mean of 78
and a variance 36.
(i) What is the probability that a student taking the examination scores higher than 72? [4]
(ii) What must be the cuttoff point for passing the examination if the examiner wants only
the top 28.1% of all scores to be passing?
[6]
4.4 A random variable X has a mean µ. = 10 and a variance a-2 = 4. Use Chebyschev's inequality
to estimate P(IX - 101~3).
[5]
END OF QUESTION PAPER
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Standard Normal Probabilities
z
z .00
-3.4
-3.3
-3.2
-3.1
-3.0
-2.9
-2.8
-2.7
-2.6
-2.5-
-2.4
-2.3
-2.2
-2.1
-2.0
-1.9
-1.8
-1.7
-1.6
-1.5
-1.4
-1.3
-1.2
-1.1
-1.0
-0.9
-0.8
-0.7
-0.6
:...0.5
-0.4
-0.3
-0.2
-0.1
-0.0
.0003
.0005
.0007
.0010
.0013
.0019
.0026
.003-5
.0047
.006i
.0082
.0107
.0139
.0179
.0228
.0287
.0359
.0446
.0548
.966~
.0808
.0968
.1151
.1357
.1587
.1841
.2119
.2420
.2743
.3085
.3446
.3821
.4207
.4602
.5000
.01
.0003
.0005
.0007
.0009
.0013
.0018
.0025
.0034
.0045
.0060
.0080
.0104
.0136
.0174
.0222
.0281
.0351
.0436
.0537
.0655
.0793
.0951
.1131
.1335
.1562
.1814
.2090
.2389
.2709
.3050
.3409
.3.783
.4168
.4562
.4960
.02
.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
.4522
.4920
Table entry for z is the area und' er the standard normal curve
to the left of z.
.03
.0003
.0004
.0006
.0009
.0012
.o6f7
.0023
.0032
.0043
.0057
.0075
.0099
.0129
.0166
.0212
.0268
.0336
.0418
.0516
.0630
.0764
.0918
.1093
.1292
.1515
.1762
.2033
.2327
.2643
.2981
.3336
.3707
.4090
.4483
.4880
.04
.0003
.0004
.0006
.0008
.0012
.0016
.0023
.0031
.0041
.0055
.0073
.0096
.0125
.0162
.0207
.0262
.0329
.M69
.0505
.0618
.0749
.0901
.1075
.1271
.1492
.1736
.2005
.2296
.2611
.2946
.3300
.3669
.4052
.4443
.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
.0015
.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
.0-087
.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
0.0
LQ.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
.1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
.J.J
3.4
.oo
.5000
.5398
.5793
.6179
.6554
.6915
.7257
.7580
.7881
.8159
.8413
.8643
.8849
.9032
.9192
.9332
.9452
.9554
.9641
.9713
.9772
.9821
.9861
.9893
.9918
.9938
.9953
.9965
.9974
.9981
.9987
.9990
.9993
.9995
.9997
z
.01
.02
.5040
· -~438
.5832
.6217
.6591
~§959
.7291
.7611
.7910
.8186
.8438
.s6tr5
.8869
.9049
.9207
.9345
.9463
.95.64
.9649
.9719
.9778
.9826
.9864
.9896
.9920
.9940
.9955
.9966
.9975
.9982
.9987
.9991
.9993
.9995
.9997
.5080
.547~
.5871
.6255
.6628
.6985
.7324
.7642
.7939
.8}12
.8461
.8686
.8888
.9066
.9222
.9357
.9474
.9573
.9656
.9726
.9783
.9830
.9868
.9~98
.9922
.9941
.9956
.9967
.9976
.9982
.9987
.9991
.9994
.9995
.9997
Tableentry for z is the area under the standard normalcurve
to the left of z.
.03
.5120
.5517
.5910
.6293
.6664
.7019
.7357
.7673
.7967
.8238
.8485
.8708
.8907
.9082
.9236
.9370
.9484
.9582
.9664
.9732
.9788
.9834
.9871
.9901
.9925
.9943
.9957
.9968
.9977
.9983
.9988
.9991
.9994
.9996
.9997
.04
.OS
.5160 .5199
.5557 .5596
.5948 .5987
.6331 .6368
.6700 .6736
.7054 .7088
.7389 .7422
.7704 .7734
.7995 .8023
.8264 .8289
.8508 .8531
.8729 .8749
.8925 .8944
.9099 .9115
.9251 .9265
.9382 .9394
.9495 .9505
.9591 .9599
.9671 .9678
.9738 .9744
.9793 .9798
.9838 .9842
.9875 .9878
.9904 .9906
.9927 .9929
.9945 .9946
.9959 .9960
.9969 .9970
.9977 .9978
.9984 .9984
.9988 .9989
.9992 .9992
.9994 .9994
.9996 .9996
.9997 .9997
.06
.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
.07
.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
.08
.5319
.5-714
.6103
.6486
.6844
.7190
.7517
.7823
.8106
.8365
.8599
.88-10
.8997
.9162
.9306
.9429
.9535
.9625
.9699
.9761
.9812
.9854
.9887
.9913
.9934
.9951
.9963
.9973
.9980
.9986
.9990
.9993
.9995
.9996
.9997
.09
.5359
.5753
.6141
.65i7
.6879
.7224
.7549
.7852
.8133
.8389
.8621
.8830
.9015
.9177
.9319
.9441
.9545
.9633
.9706
.9767
.9817
.9857
.9890
.9916
.9936
.9952
.9964
.9974
.9981
.9986
.9990
.9993
.9995
.9997
.9998