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PBT501S - PROBABILITY THEORY 1 - 1ST OPP - NOV 2022 |
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n Am I BI A u n IVER s I TY
OF SCIEnCE Ano TECHn0L0GY
FACULTY OF HEAL TH, APPLIED SCIENCES, AND NATURAL
RESOURCES
DEPARTMENTOF MATHEMATICSAND STATISTICS
QUALIFICATION: BACHELOROF SCIENCE
QUALIFICATIONCODE: 07BOSC
COURSE:PROBABILITY THEORY 1
DATE:NOVEMBER 2022
DURATION: 3 HOURS
LEVEL:5
COURSECODE:PBT501 S
SESSION: NOVEMBER
MARKS: 100
EXAMINER(S)
FIRSTOPPORTUNITYEXAMINATION QUESTIONPAPER
Dr. D. Ntirampeba
MODERATOR: Mr. J. Amunyela
THIS QUESTIONPAPERCONSISTSOF 4 PAGES
(Excluding this front page and statistical tables)
INSTRUCTIONS
1. Answer ALLthe questions.
2. Write clearly and neatly.
3. Number the answers clearly.
PERMISSIBLEMATERIALS
1. Non-programable calculator
ATTACHMENTS
1. Statistical tables (Z-Table)
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QUESTION 1 [40 Marks]
1.1. Define the following terminologies as they are applied in set theory and probability theory
1.1.1 A random experiment
[2]
1.1.2 A sample space
[2]
1.1.3 The size of a set A
[2]
1.1.4 Pairwise mutually exclusive events
[2]
1.1.5 Independent events (say A and B)
[2]
1.2. The probability mass function of the discrete random variable X is given by
p (x) = {c(;) (5~x), x = 0,1,2,3,4
0, elsewhere
1.2.1. Find the value of c
[2]
= 1.2.2. Assuming c -2...f.i,nd the cumulative distribution of X
[4]
126
1.2.3. Find the median
[1]
1.3. An important factor in solid missile fuel is the particle size distribution. Significant
problems occur if the particle sizes are too large. From production data in the past, it
has been determined that the particle size (in micrometers) distribution is characterized
by
kx- 4, x > l
f (x) = { 0, otherwise
1.3.1. Find the value of k so that f (x) is a probability density function.
[3]
1.3.2. Assuming the value of k is 3, find:
1.3.2.1. the expected number of X
[3]
1.3.2.2. the variance of number of X
[4]
1.3.2.3. What is the probability that a random paiiicle from the manufactured fuel exceeds 4
micrometers
[3]
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1.4. 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 IO
1234
f(x) 0.41 0.37 0.16 0.05 0.01
Work out the following.
1.4.1. standard deviation of the number of imperfections
[3]
1.4.2. E(X 2 + 3)
[2]
1.4.3 Var(X 2 + 3)
[5]
QUESTION 2 [20 Marks]
2.1. At Hopewell Electronics, all 140 employees were asked about their political affiliations.
The employees were grouped by type of work, as executives or production workers. The
results are shown in the table below.
Employee type
Executive
Production
worker
Democrat
5
Political affiliation
Republican Independent
34
9
63
21
8
Suppose an employee is selected at random from Hopewell employees. Let us use the
following notations to represent different events of choosing:
E =executive; W =production worker; D =democrat; R =republican; and I =independent
2.1.1 ComputeP(R)
[1]
2.1.2 ComputeP(D and E)
[2]
2.1.3 Compute P(W or I)
[2]
2.1.4 ComputeP(EID)
[3]
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2.2.
A paint-store chain produces and sells latex and semigloss paint. Based on long-range sales,
the probability that a customer will purchase latex paint is 0.75. Of those that purchase latex
paint, 60% also purchase rollers. But only 30% of semigloss paint buyers purchase rollers. A
randomly selected buyer purchases a roller and a can of paint. What is the probability that
the paint is latex?
[5]
2.3.
Customers who purchase a certain make of car can order an engine in any of three sizes. Of all
cars sold, 45% have smallest engine, 35% have the medium size one, and 20% have largest. Of
cars with small engine, 10% fail an emissions test within two years of purchase, while 12% of
those with medium size and 15% of those with largest engine fail. A record for a failed
emissions test is chosen at random.
2.3.1 What is probability that it is for a car with medium-size engine?
[4]
2.3.2. What is probability that it is for a car with medium-size engine or largest engine? [3]
QUESTION 3 [20 Marks]
3.1. A soft-drink machine is regulated so that it discharges an average of 200 milliliters per
cup. If the amount of drink is normally distributed with a standard deviation equal to
15 milliliters,
3.1.1. what fraction of the cups will contain more than 224 milliliters?
[4]
3.1.2. below what value do we get the smallest 25% of the drinks?
[5]
3.2. In a certain city district, the need for money to buy drugs is stated as the reason for
75% of all thefts.
3.2.1. Find the probability that among the next 5 theft cases reported in this district, at most
3 resulted from the need for money to buy drugs.
[4]
3.2.2. What is the expected number of theft cases resulting from the need for money to buy
drugs if 10 theft cases are observed?
[2]
3.3. A secretary makes 2 errors per page, on average.
3.3.1. What is the probability that on the page he or she will make 4 or more errors? [4]
3.4.2. What is the expected number of errors in the next three pages?
[1]
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QUESTION 4(20 Marks]
4.1. A random variable X has a meanµ = 10 and a variance <Y2 = 4. Use Chebyshev's
theorem find
P(S < X < 15)
[5]
4.2 The length of time for one individual to be served at a cafeteria is a random variable
having an exponential distribution with a mean of 4 minutes.
4.2.1. What is the probability that a person is served in less than 3 minutes?
[5]
4.2.2. Find the variance of the length of time for an individual to be served at the cafeteria.
[2]
4. 3.
4.3.1. Let X be continuous uniform random variable on the interval [a, b]. show that
E(X) = a+b
[4]
2
4.3.2. The daily amount of coffee, in liters, dispensed by a machine located in an airport
lobby is a random variable X having a continuous uniform distribution in interval
[7, 10]. Find the probability that on a given day the amount of coffee dispensed by
this machine will be at most 8.8 liters?
[4]
END OF EXAMINATION QUESTION PAPER
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Standard Normal Probabilities
Table entry for z is the area under the standard normal curve
z
to the left of z.
z
.00
.01
.02
.0003 .0003 .0003
. .QOOJ
.0006
.0009
.0013 .0013 .0013
.0019 ""]o18 .0018
-2.8 .0026 .0025 .0024
-2.7 .0035 .0034 .0-633
-2.6
.0045 .0044
.0059
-2.4 .0082 .0080 .0078
-2.3 .. 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 .0427
-1.6 .0548 .0537 .0526
-1.5 .0608 .0655 .064)
-1.4 .0808 .0793 .0778
-1.3 .0968 .0951 .0934
-1.2 .1151 .1131 .1112
-1.1 -lJ57 .. 13~5 .J314
-1.0 .1587 .1562 .1539
-0.9 .1841 .1814 .1788
-0.8 .2119 .2090 .2061
-0.7 .242,0 .2389 .2358
-0.6 .2743 .2709 .2676
-0.5 -l013.? .3Q~Q. .301_5
-0.4 .3446 .3409 .3372
-0.3 .3821 .3783 .3745
-0.2 .4207 .4168 .4129
-0.1 .4602 .4562 .4522
-0.0 .5000 .4960 .4920
.03
.0003
._9004
.0006
.0009
.0012
:0017
.0023
~0032
.0043
.0057
.0075
.0099
.0129
.0166
.0212
.0268
.0336
.0418
.0516
.~o
.0764
.0918
.1093
.129~
.1515
.1762
.2033
.2327
.2643
-~81
.3336
.3707
.4090
.4483
.4880
.04
.0003
.0004
.0006
.0008
.0012
.0016
.0023
.0031
.0041
.0055
.0073
.0096
.0125
.0162
.0207
.0262
.0329
.0409
.0505
.0618
.0749
.0901
.1075
,!271
.1492
.1736
.2005
.2296
.2611
.2946
.3300
.3669
.4052
.4443
.4840
.05
.06
.07
.08
.0003 .0003 .0003 .0003
.OOQ4 .,:0004 .0004 .0004
.0006 .0006 .0005 .0005
.0008 .0008 .ocRis .0007
.0011 .0011 .0011 .0010
.0016 .0015. .0015 .0014
.0022 .0021 .0021 .0020
.0030
.0040 .0039 .0038 .0037
- .0054 .0052 .0051 .0049
.0071 .0069 .0068 .0066
.0094 .0091 .0089 .0087
.0122 .0119 .0116 .0113
.0158 .0154 .0150 .0146
.0202 .0197 .0192 .0188
.0256 .0250 .0244 .0239
.0322 .0314 .0307 .0301
.0401 .0392 .0384 .0375
.0495 .0485 .0475 .0465
.O~Q§ .0594 .0582 .0571
.0735 .0721 .0708 .0694
.0885 .0869 .0853 .0838
.1056 .1038 .1020 .1003
.1251 .1230 .1210 .1190
.1469 .1446 .1423 .1401
.1711 .1685 .1666 .1635
.1977 .1949 .1922 .1894
.2266
.2206 .2177
.2578 .2546 .2514 .2483
.2~1~ .
.2843 !.28!0
.3264 .3228 .3192 .3156
.3632 .3594 .3557 .3520
.4013 .3974 .3936
.44()4 .4364
.4801 .4761 .4721 .4681
.09
.0002
.0003
.0005
.0007
.0010
.0014
.0036
.0048
.0064
.oo-84';
.0110
.0143
.0183
.0233
.0294
.0367'
.0455
.05i2_
.0681
.0823
.0985
.1170
.1379
.1611 ·
.1867
.2148
.2451
.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 .00
.01
.02
.03
.04
.05
.06
.07
.08
.09
0.0 .5000 .5040 .5080 .5120 .5160 .5199
r;OJ .5,398'. .5438 .5478 :~~.F .5557 .5596
.5279 .5319 .5359
.5714 .5?,?,}
0.2 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .6141
.6217 .6255 · .6_293 .6331 .6368 .6406 .6.443 .6480 .6517
.6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .6879
.7257
.7580
.7881
-.:7n2,1911--
.7324
.7642.
.7910 .7939
.7357 .7389 .7422 .7454 .7486 .7517
.7673 .710,r f:734 .7764. _179·4 .7823
.7967 .7995 .8023 .8051 .8078 .8106'
.7549
.7852
.8133
.81§6 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389
1.0 .8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .8621
.8665 .•.. 8686 .870-8 .8729 .8749 .877() .8790 :a810 .8830
w
•
-
1.2
.8869
.8907 .8925
.8962 .8980 .8997 .9015
.9099 -
1.4 .9192 .9207 .9222 .9236 .9251 .9265 .9279
.9162
.9306
.9177'
.9319
1.5 .9332 .9345 .9357 ..93-70 .9382 .9394 .9406 .9418 .9429 .9441'
1.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545
.9573 .9582_ .9591 .9599 .9608 .9616 .9625 .9,[33
.9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9706
.97J.§. .:2ZR- .9738 .9744 .9750 .9756 .9761
2.0 .9772 .9778 .9783 .9788 .9793 .9798 .9803
.9812 .9817
,2.1 .9821 .9826 .9830 .9834 .9838
2.2 .9861 .9864 .9868 .9871 .9875 .9878
.9884 .9887 .9890
2.3 .9893 .9896 -~898 .9901 .99Q4 .9906
.9911 .9913 .9916
2.4
.9920 .9922 .9925 .9927 .9929 .9931 .9932 .9934 .9936
.9941 7943
.9945 .9946 .9948 .9.949 . .9951 .9952
2.6 .9953 .9955 .9956 .9957 .9959 .9960 .9961 .9962 .9963 .9964
·2.7 .9965 .9966 .9967 ,9968 .9969 .9970 .9971 .9972 .9973 .9974
2.8 .9974
.9977 .9977 .9978 .9979 .9979 .9980
.9984
.9987
.9988
.9989 .9989
.9990 .9990
.9991
.9992 .9992
.9993 .99937
3.2
3.3
.9993
.9995
.9993
.9995
.9994 .9994
.9995 . .9996
.9994
.9996
.9994
.9996
.9994 .9995 .
.9996 ...:.9~f96
.9995
.99%
.9995
3.4 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9998