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PBT501S- PROBABILITY THEORY 1- JAN 2020 |
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NAMIBIA UNIVERSITY
OF SCIENCE AND TECHNOLOGY
FACULTY OF HEALTH AND APPLIED SCIENCES
DEPARTMENT OF MATHEMATICS AND STATISTICS
QUALIFICATION: Bachelor of Science ; Bachelor of Science in Applied Mathematics and Statistics
QUALIFICATION CODE: 07BOSC
LEVEL: 5
COURSE CODE: PBT501S
COURSE NAME: PROBABILITY THEORY 1
SESSION: JANUARY 2020
DURATION: 3 HOURS
PAPER: THEORY
MARKS: 90
SECOND OPPORTUNITY / SUPPLEMENTARY EXAMINATION QUESTION PAPER
EXAMINER
Mr ROUX, A.J
MODERATOR:
Dr NTIRAMPEBA, D
INSTRUCTIONS
1. Answer ALL the questions in the booklet provided.
Show clearly all the steps used in the calculations.
3. All written work must be done in blue or black ink and sketches must
be done in pencil.
PERMISSIBLE MATERIALS
1. Non-programmable calculator without a cover.
ATTACHMENTS
1. Standard Normal Probability Table
THIS QUESTION PAPER CONSISTS OF 4 PAGES (Including this front page)
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QUESTION 1
[18]
1.1) Carefully distinguish between a discrete and a continuous random variable,
and give an example of each.
(2x2 =4)
1.2) Define the following terminologies as they are applied to the Probability
Theory.
1.2.1) A probability mass function
(3)
1.2.2) A probability density function
(3)
1.3) The records of a department store show that 20% of their customers who
make a purchase return the merchandise in order to exchange it.
1.3.1) What is the probability that in the next eight purchases, at least two
customers return the merchandise for exchange?
(4)
1.3.2) Find the mean and the standard deviation of the number of customers who
make a purchase return the merchandise in order to exchange it.
(4)
QUESTION 2
[25]
The probability mass function of a discrete random variable X is given by
n(x) = =
x=2,4,6,8
0,
otherwise
2.1) Find the value of a constant k
(3)
2.2) Construct a probability mass table for X
(3)
2.3) Find cumulative probability of X
(2)
2.4) Find P(2<X <5)
(2)
2.5) Find the mean ofX
(4)
2.6) Find the variance and standard deviation of X
(8)
2.7) Find the Coefficient of variation X
(3)
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QUESTION 3
[15]
The Office of Student Records has shown that 7% of student accounts are paid in
full at registration. The Office of Bursar has just randomly selected accounts of 300
students from this college to find the probability that:
3.1) exactly 28 student accounts are paid in full at registration
(7)
3.2) under 25 student accounts are paid in full at registration
(4)
3.3) more than 18 student accounts are paid in full at registration
(4)
QUESTION 4
[32]
4.1) Acaris randomly chosen among all those passing through the Brakwater
Police Road Block on a certain day. The probability that the car is yellow is
3/100 , the probability that the driver is blonde is 1/5 , and the probability
that the car is yellow and the driver is blonde is 1 / 50. Find the conditional
probability that the driver is blonde given that the car is yellow.
(4)
4.2) In amultiple choice question, there are five different answers, of which only
one is correct. The probability that a student will know the correct answer is
0.6. If a student does not know the answer, he/she guesses an answer.
4.2.1) What is the probability that the student gives the correct answer?
(3)
4.2.2) lf the student gives the correct answer, what is the probability that he
guessed?
(5)
4.3) Five different lifeguards are available for duty one Saturday afternoon. There
are three lifeguard stations. In how many ways can the three life guards can
be chosen and arranged among the three lifeguard stations.
(3)
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4.4) Consider the events A = {1,3,4}, B = {2,5,7}, and C = {6,7,8} over a possibility
space S = {1,2,3,4,5,6,7,8}. With reasons, state whether true or false:
4.4.1 Events A, B, and C are mutually exhaustive.
[3]
4.42. A, B, and C are partitioning events.
[3]
4.5) How many different ways can we arrange the letters ABEACDEAACCB? [3]
4.6) We are making number plates for cars. Suppose that our valid number plate
must have exactly two letters and exactly 6 digits such that all the digits lie
between the letters and that a zero (0) digit is not allowed to appear first. How
many different number plates are possible if:
4.6.1) repetitions are allowed for both letters and digits?
[4]
4.5.2) repetitions are NOT allowed for both letters and digits?
[4]
XXXXXXXXXXXKXXXXKXXXXXXXXX XXXXKK KKXKXX AXKK XKXKXKXKXXKXXKXKXXX X KXKXXXX XXXKKXXXXXXKKX XXKK
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Standard Normal Cumulative Probability Table
Cumulative probabilities for NEGATIVE z-values are shown in the following table:
a aAS
Zz
3.4
3.3
“3.2
“3.4
3.0
“2.9
“2.8
“27
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
“1.4
-1.0
-0.9
0.3
0.7
“0.6
“0.5
6.4
0.3
“0.2
0.4
O.0
0.00
0.0003
0.0005
0.0007
0.0010
0.0013
0.0019
0.0026
0.0035
0.0047
0.0062
0.0082
0.0107
0.0139
0.0179
0.0228
0.0287
0.0359
0.0446
0.0548
0.0668
0.0808
0.0968
0.1151
0.1357
0.1587
0.1844
0.2119
0.2420
0.2743
0.3085
0.3446
0.3827
0.4207
0.4602
0.5000
0.04
0.0003
0.0005-
0.0007
0.0009
0.0013
0.02
0.0003
0.0005
0.0006
0.0009
0.0013
0.0018
0.0025
0.0034
0.0045
0.0080
0.0018
0.0024
0.0033
0.0044
0.0059
0.0080
0.0104
0.0136
0.0174
0.0222
0.0078
0.0102
0.0132
0.0170
0.0217
0.0281
0.0351
0.0436
0.0537
0.0655
0.0274
0.0344
0.0427
0.0526
0.0643
0.0793
0.0951
0.1131
0.1335
0.1562
0.0778
0.0934
0.1112
0.1314 -
0.1539 -
0.1814
0.2090
0.2389
0.2709
0.3050
0.1788
0.2061
0.2358
0.2676
0.3015
0.3409
0.3783
0.4168
0.4562
0.4960
0.3372°
0.3745
~ 0.4129
0.4522
0.4920
0.03
0.0003
0.0004.
0.0006
0.0009
0.0012
0.04
0.0003
0.0004
0.0006
0.0008
0.0012
0.05
0.0003
0.0004°
0.0006
0.0008
0.004 1.
0.06
0.0003
0.0004
0.0006
0.0008
0.0011
0.07
0.0003
0.0004
0.0005
0.0008
0.0011
0.08
0.0003
0.0004
0.0005
0.0007
0.0010
0.09
0.0002
0.0003
0.0005
0.0007
0.0010
0.0017
0.0023
0.0032
0.0043
0.0057
0.0016
0.0023
0.0031
0.0041
0.0055
0.0016
0.0022
0.0030
0.0040
0.0054
0.0015
0.0021
0.0029
0.0039
0.0052
0.0015
0.0021
0.0028
0.0038
0.0051
0.0014
0.0020
0.0027
0.0037
0.0049
0.0014
0.0019
0.0026
0.0036
0.0048
0.0075
0.0099
0.0129
0.0166
0.0212
.0.0073 - 0.0071
0.0069
0.0096
0.0094
0.0091
0.0125
0.0122
0.0119
0.0162
0.0158
0.0154
0.0207
0.0202 ~ 0.0197
0.0068
0.0089
0.0116
0.0150
0.0192
0.0066
0.0087
0.0113
. 0.0146
0.0188
0.0064
0.0084
0.0110
0.0143
0.0183
0.0268
0.0336
0.0418
0.0516
0.0630
0.0262
0.0329
0.0409"
0.0505
0.0618
0.0256
0.0322
0.0401
0.0495
0.0606
0.0250
0.0314
0.0392
0.0485
0.0594
0.0244 ° 0.0239
0.0307
0.0301
- 0.0384
0.0375
0.0475
0.0465
0.0582 - 0.0571
0.0233
0.0294
0.0367
0.0455
0.0559
0.0764
-0.0918
0.1093
0.1292
0.1515
0.0749
0.0901
0.1075
0.1271
0.1492
0.0735
0.0885
0.1056
0.1251
0.1469
0.0721
0.0869
0.1038
0.1230
0.1446
0.0708
0.0853
0.1020
0.1210
0.1423
0.0694
0.0838
0.1003
0.1790
0.1401
0.0681
0.0823
0.0985
0.1170
0.1379
0.1762
0.2033
0.2327
0.2643
0.2981
0.1736
0.2005
0.2296
0.2614
0.2946
0.1711
0.1977
0.2266
0.2578
0.2912
0.1685
0.1949
0.2236
(0.2546
0.2877
0.1660
0.1922
0.2206
0.2514
0.2843
0.1635
0.1894
0.2177
0.2483
0.2810
0.1614
0.1867
0.2148
0.2451
0.2776
0.3336
0.3707
0.4090
0.4483
0.4880
0.3300
0.3669
0.4052
0.4443
0.4840
0.3264
0.3632
0.4013
0.4404
0.4801
0.3228
0.3594
0.3974
0.4364
0.4761
0.3192
0.3557
. 0.3936
0.4325
0.47214
0.3156
0.3520
0.3897
0.4286
0.4681
0.3121
-0,3483
0.3859
0.4247 ~
0.4644
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Standard Normal Cumulative Probability Table
Cumulative probabilities for POSITIVE z-values are shown in the following table:
Zz
0.00
0.01
0.02
0.63
0.04
6.05
0.06
0.07
6.08
0.09
8.0
_ 0.5000
0.5040
0.5080
0.5120
0.5160
0.5199
0.5239
0.5279
0.5319
0.5359
0.1
0.5398
0.5438
0.5478
0.5517
0.5557
0.5596 ~ 0.5636
0.5675
0.5714
0.5753
0.2
0.5793
0.5832
0.5871
0.5910
0.5948
0.5987
0.6026
0.6064
0.6103
0.6141
0.3
0.6179
0.6217
0.6255
0.6293
0.6331
0.6368
0.6406
0.6443
0.6480
0.6517
0.4
0.6554
0.6591
0.6628
0.6664
0.6700
0.6736
0.6772
0.6808
0.6844
0.6879
0.5
0.6915
0.6950
0.6985 0.7019
0.7054
0.7088
0.7123
0.7157 0.7190
0.7224
0.6
0.7257
0.7291
0.7324 0.7357
0.7389
0.7422
0.7454
0.7486
0.7517
0.7549
0.7
0.7580
0.7611
0.7642 0.7673
0.7704 = 0.7734
0.7764
0.7794
0.7823
0.7852
0.8
0.7881
0.7910
0.7939
0.7967
0.7995
0.8023
0.8051
0.8078
0.8106
0.8133
0.9
0.8159
0.8186
0.8212
0.8238
0.8264
0.8289 . 0.831%
0.8340
0.8365
0.8389
1.0
0.8413
0.8438
1.1
0.8643
0.8665
1.20
0.8849
0.8869
1.3.
0.9032 ~ 0.9049
1.4 | 0.9192
0.9207
0.8461
0.8686
0:8888
0.9066 -
0.9222 .
0.8485
0.8708
0.8907
0.9082
0.9236
0.8508
0.8729
0.8925
0.9099
0.9251
0.8531
0.8749
0.8944
0.9115
0.9265
0.8554
0.8770
0.8962
0.9131
0.9279
0.8577
0.8790
0.8980
0.9147
0.9292
0.8599
~- 0.8810
0.8997
0.9762
0.9306
0.8621
0.8830
0.9015
0.9177
0.9319
1.5
0.9332
0.9345
0.9357
0.9370
0.9382
0.9394
0.9406
0.9418
0.9429
0.9441
1.6
0.9452
0.9463
0.9474
0.9484
0.9495
0.9505
0.9515
0.9525
0.9535
0.9545
1.7
0.9554 ° 0.9564 . 0.9573 0.9582
0.9591
0.9599
0.9608
0.9616
0.9625
0.9633
1.8
0.9641 0.9649
0:9656 .0.9664 - 0.9671
0.9678
0.9686 _ 0.9693 0.9699
0.9706
1.9
0.9713
0.9719
0.9726
0.9732
0.9738
0.9744
0.9750
0.9756
0.9761
0.9767
2.0
0.9772
0.9778
0.9783
0.9788
0.9793
0.9798
0.9803
0.9808
0.9812
0.9817
2.1
.0.9821.. 0.9826
0.9830
0.9834
0.9838
0.9842
0.9846
0.9850
0.9854
0.9857
2.2
. 0.9861
0.9864
0.9868
0.9871
0.9875
0.9878
0.9881
0.9884
0.9887, 0.9890
2.3
0.9893
0.9896
0.9898
0.9901
0.9904
0.9906
0.9909
0.9911
0.9913
0.9916
2.4
0.9918 0.9920
0.9922 0.9925
0.9927
0.9929
0.9931
0.9932
0.9934
0.9936
2.5
0.9938
0.9940
0.9944
0.9943
0.9945
0.9946
0.9948
0.9949
0.9951
0.9952
2.6
0.9953
0.9955
0.9956
0.9957
0.9959 0.9960
0.9961
0.9962
0.9963
0.9964
2.0
0.9965
0.9966
0.9967
0.9968
0.9969
0.9970
0.9971
0.9972
0.9973
0.9974
2.3
0.9974
0.9975
0.9976
0.9977
0.9977 0.9978
0.9979
0.9979
0.9980
0.9981
2.9
0.9981
0.9982 0.9982 0.9983
0.9984
0.9984
0.9985
0.9985
0.9986 0.9986
3.0
0.9987
0.9987
0.9987
0.9988
0.9988
0.9989
0.9989
0.9989
0.9990
0.9990
3.1
0.9990
0.9991
0.9991
0.9991
0.9992 0.9992
0.9992
0.9992
0.9993
0.9993
3.2
0.9993
0.9993
0.9994
0.9994
0.9994. 0.9994
0.9994
0.9995 ~ 0.9995
0.9995
3.3
0.9995
0.9995 0.9995
0.9996 ~ 0.9996
0.9996
0.9996
0.9996
0.9996
0.9997
3.4.
0.9997
0.9997
0.9997
0.9997
0.9997
0.9997
0.9997
0.9997
0.9997
0.9998