Fall 2015
Math 141:505
Exam 3
Form A
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this academic work.
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INSTRUCTIONS
• Part 1: Multiple Choice (Problems 1-8) Each multiple choice problem is worth 6 points for a total of 48 points.
Answers should be written in the boxes provided. No partial credit will be given.
• Part 2: Work Out (Problems 9-13) The number of points for each problem is indicated next to the problem for
a total of 52 points. Partial credit will be given. All steps must be written clearly and neatly to receive credit.
If you use your calculator for anything beyond an arithmetic calculation, please indicate how at the appropriate
step. Box your final answer.
“An Aggie does not lie, cheat or steal or tolerate those who do.”
c
Yeong-Chyuan
Chung, November 20, 2015
Part 1: Multiple-Choice. Each multiple choice problem is worth 6 points for a total of 48 points. No
partial credit will be given. Answers should be written in the boxes provided.
1. An experiment consists of selecting a card at random from a deck of 52 cards. What is the probability that a
heart or an ace is drawn?
(a)
(b)
(c)
(d)
17
52
16
52
8
52
1
52
b
2. If E and F are independent events, P(E) = .45, and P(F) = .5, what is P(E c ∪ F c )?
(a) .775
(b) .725
(c) .275
(d) .225
a
3. Which of the following is an infinite discrete random variable?
(a) The number of heads that occur when a coin is tossed five times.
(b) The distance (in miles) a commuter travels to work.
(c) The number of boys in a two-child family.
(d) The number of times a die is rolled until a 6 falls uppermost.
d
4. If the odds in favor of a team winning a particular football match are 7 to 5, what is the probability that the team
will win the match?
(a)
(b)
(c)
(d)
5
12
5
7
7
12
7
5
c
2
c
Yeong-Chyuan
Chung, November 20, 2015
5. Two light bulbs are selected at random from a lot of 10, of which 4 are defective. What is the probability that
both light bulbs are defective? (Answers below are rounded to two decimal places.)
(a)
(b)
(c)
(d)
.03
.13
.16
.21
b
6. The personnel department of a company compiled the following data regarding the income and education of its
employees:
Income $65,000 or below Income above $65,000
Noncollege graduate
2040
840
College graduate
400
720
What is the probability that a randomly chosen employee has income above $65,000 if it is known that he or she
has a college degree?
(a)
(b)
(c)
(d)
720
4000
1560
4000
720
1560
720
1120
d
7. Refer to the table in the previous problem. What is the probability that a randomly chosen employee has income
above $65,000 or is a college graduate?
(a)
(b)
(c)
(d)
720
4000
1240
4000
1960
4000
3160
4000
c
8. A mathematics test consists of eight multiple-choice questions. If each question has four possible answers, of
which only one is correct, what is the probability that a student who guesses at random on each question will
answer at most two questions correctly? (Answers below are rounded to two decimal places.)
(a)
(b)
(c)
(d)
.31
.37
.59
.68
d
3
c
Yeong-Chyuan
Chung, November 20, 2015
Part 2: Work Out. The number of points for each problem is indicated next to the problem. Partial credit will
be given. All steps must be written clearly and neatly to receive credit. If you use your calculator for anything
beyond an arithmetic calculation, please indicate how at the appropriate step. Box your final answer.
9. (8pts) Let S = {s1 , s2 , s3 , s4 , s5 , s6 } be the sample space associated with an experiment having the following
probability distribution:
Outcome
Probability
(a) Find P({s1 , s3 , s5 }).
Solution:
P({s1 , s3 , s5 }) = P(s1 ) + P(s3 ) + P(s5 ) =
1
12
s1
s2
a
1
12
s3
1
12
s4
b
s5
s6
1
3
1
12
1
+ 12
+ 13 = 12 .
7
(b) If P({s3 , s4 , s5 }) = 12
, find a and b.
Solution:
7
12 = P({s3 , s4 , s5 }) = P(s3 ) + P(s4 ) + P(s5 ) =
1
12
+ b + 13 =
5
12
+ b so b =
1
1
1
All the probabilities should sum to 1, so a = 1 − 12
− 12
− b − 13 − 12
=
4
3
12
7
12
5
− 12
=
= 14 .
2
12
= 16 .
c
Yeong-Chyuan
Chung, November 20, 2015
10. (10pts) The relative humidity, in percent, in the morning for the months of January through December in Boston
follows:
70, 68, 67, 69, 69, 71, 73, 74, 76, 79, 77, 74.
Find the
(a) mean,
72.25 (Can use 1-variable stats, or just compute manually.)
(b) median,
72 (First, rearrange the list in either increasing or decreasing order. Since there is an even number of data
points, there are two in the middle. Take the average of the middle two.)
(c) mode(s),
69 and 74
(d) standard deviation,
3.677068579 (Use 1-variable stats.)
(e) variance
(3.677068579)2 = 13.5208
of this set of data.
11. (10pts) Which of the following events is more likely to occur? Justify your answer.
E: Getting a six at least once in 4 throws of a single fair die.
F: Getting a double six at least once in 24 throws of a pair of fair dice.
Solution:
P(E) = 1 − binompd f (4, 61 , 0) ≈ .5177
1
P(F) = 1 − binompd f (24, 36
, 0) ≈ .4914
Event E is more likely.
5
c
Yeong-Chyuan
Chung, November 20, 2015
12. (14pts) The chief loan officer of La Crosse Home Mortgage Company summarized the housing loans extended
by the company in 2014 according to type and term of the loan. Her list shows that 70% of the loans were fixedrate mortgages (F), 25% were adjustable-rate mortgages (A), and 5% belong to some other category (O). Of the
fixed-rate mortgages, 80% were 30-year loans and 20% were 15-year loans; of the adjustable-rate mortgages,
40% were 30-year loans and 60% were 15-year loans; finally, of the other loans extended, 30% were 15-year
loans, 60% were 10-year loans, and 10% were for a term of 5 years or less.
(a) Draw a tree diagram representing the data.
Solution:
.8
F
..
...
...
...
.
...
...
..
..
.
.
.
...
...
...
..
.
.
...
...
...
..
.
.
...
...
...
..
.
...
......................................
..
........
........
........
........
30
.2
........
........
........
........
...
15
.7
.25
..
.4...........................
30
....
......
A
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
.
......
......
......
......
......
......
....
.6
15
.05
15
.3..........................
.
......
......
.6
......................................
10
O .........
...... .1
......
......
......
......
.
≤5
(b) What is the probability that a home loan extended by La Crosse has an adjustable rate and is for a term of
15 years?
Solution:
P(A ∩ 15) = (.25)(.6) = .15
(c) What is the probability that a home loan extended by La Crosse is for a term of 30 years?
Solution:
P(30) = (.7)(.8) + (.25)(.4) = .66
(d) What is the probability that a 15-year loan has a fixed rate?
Solution:
P(F|15) =
P(F∩15)
P(15)
=
(.7)(.2)
(.7)(.2)+(.25)(.6)+(.05)(.3)
=
28
61
6
≈ .459
c
Yeong-Chyuan
Chung, November 20, 2015
13. (10pts) Four balls are selected at random without replacement from an urn containing four green balls, two red
balls, and two blue balls. Let the random variable X denote the number of blue balls drawn.
(a) Find the probability distribution of the random variable X.
Solution:
x
P(X = x)
0
C(2,0)C(6,4)
C(8,4)
1
=
15
70
C(2,1)C(6,3)
C(8,4)
2
=
40
70
C(2,2)C(6,2)
C(8,4)
=
15
70
(b) Compute E(X).
Solution:
40
15
E(X) = 0( 15
70 ) + 1( 70 ) + 2( 70 ) = 1
Problem Part 1
9
10
11
12
Score
7
13
Total