Final Exam Review F09 O’Brien
FM Lial 9th
Finite Mathematics Final Exam Review
Read and carefully follow all directions. Show all of your work on every problem. Work the problems on blank paper
with no resources other than the departmental formula sheet and your calculator. If you have to peek at something
(homework, notes, textbook, solutions manual, etc.) to complete a problem, take note of what knowledge you were missing,
and work some additional problems of this type.
To prepare for questions from 3.1 – 7.6, rework Exams 1 and 2. If you have difficulty with a particular problem, work
additional problems of that type from the Exam 1 Review and/or the Exam 2 Review and/or the homework assignments.
In addition to this review, you should work the Departmental Final Exam Review.
8.1
The Multiplication Principle; Permutations
1.
A restaurant menu lists 6 appetizers, 10 entrees, and 5 desserts. How many ways can a diner select a three-course
meal?
2a.
In how many ways can a club with 8 members elect three officers – a president, a vice-president, and a treasurer?
2b.
A baseball team has 15 players. How many 9-player batting orders are possible?
3.
Nine books are to be arranged on a shelf. Four of the books have red covers, 3 have green covers, and 2 have gray
covers. In how many ways can the books be arranged on a shelf if books of the same color must be grouped together?
4.
How many four-letter radio station call letters are there if
a.
Each must begin with a K or a W?
b.
Each must begin with a K or a W and no letter may be repeated?
c.
Each must begin with a K or a W, no letter may be repeated, and the last letter must be a Z?
5.
How many distinguishable permutations are there of the letters in the word initial?
8.2
Combinations
6.
Three teachers are to be selected from a group of 28 to work on a special project.
a.
In how many ways can this group of three be selected?
b.
In how many ways can the group of three be selected if one particular teacher must be in the group?
c.
In how many ways can a group of at most 3 teachers be selected?
7a.
Use a tree diagram to find the number of ways 2 letters can be chosen from the set {a, b, c} if order matters and
repetition is allowed.
b.
Reconsider part a if no repeats are allowed.
8.
For each problem, decide whether permutations or combinations should be used, and then solve the problem.
a.
How many 3 digit security codes are possible if no digit may be repeated?
b.
XYZ Corporation has decided to form a secretarial pool to serve its four chief executive officers. In how many ways
can four secretaries be selected from a group of twenty?
c.
Five cards are dealt from a standard deck of 52 cards. How many such hands have only face cards?
d.
In how many ways can the names of seven candidates running for the same office be listed on a ballot?
9.
From a group of 15 smokers and 21 nonsmokers, a researcher must select 13 individuals to participate in a research
study. In how many ways can this be done if the study group must contain exactly 8 smokers?
1
FM Lial 9th
Final Exam Review F09 O’Brien
Whenever possible, give the exact form of a probability (i.e., a reduced fraction) followed by a decimal approximation
rounded to 3 decimal places.
8.3
Probability Applications of Counting Principles
10.
The Benton County Health Inspector is considering inspecting 6 restaurants for health violations. Three are in
Bentonville, two are in Rogers, and one is in Centerton. He decides to inspect 2 restaurants, chosen at random, 1 this
month and 1 next month. If each restaurant is equally likely to be chosen and no restaurant can be inspected twice,
what is the probability that 1 restaurant in Rogers and 1 restaurant in Centerton are selected?
11.
Among 12 microwave ovens, 10 are in working order and two are defective. What is the probability that a sample
of five, selected at random, without replacement and without regard to order, contains:
a.
5 in working order?
b.
exactly 2 that are not in working order?
c.
at least 4 are in working order?
12.
What is the probability of getting a full house (3 cards of one value and 2 of another) in 5-card poker?
13.
A political action committee contains 3 female Republicans, 4 male Republicans, 5 female Democrats, 2 male
Democrats; 6 female members of the Green Party, and 1 male member of the Green Party. The group must select a
delegation of 5 of its members to speak at a forum. Find the probability that the delegation contains:
a.
exactly 2 Democrats
b.
no more than 3 Republicans
c.
more women than men
14.
What is the probability that at least 2 students in a class of 30 have the same birthday? Give your answer in decimal
form, rounded to 4 decimal places.
8.4
Binomial Probability
15.
A single fair die is rolled 5 times. Find the probability of rolling:
a.
all threes
b.
no threes
c.
exactly 2 threes
d.
at least 2 threes
16.
A factory tests a random sample of 20 transistors for defects. The probability that a particular transistor will be
defective has been established by past experience to be .05. Find the probability that the number of defective
transistors in the sample is at most 2. Give your answer in decimal form, rounded to 3 decimal places.
17.
The probability that a male will be color-blind is .042. Find the probability that in a group of 53 men at least one, but
no more than four, men are color-blind. Give your answer in decimal form, rounded to 3 decimal places.
8.5
Probability Distributions; Expected Value
18a.
Prepare a probability distribution and a histogram for the following experiment. Let x be the random variable.
Three balls are drawn from a bag in which there are 6 red balls & 4 black balls and the number of red balls is counted.
b.
On the histogram, shade the region which represents P(x ≤ 2).
c.
What is the expected number of red balls in the sample?
2
Final Exam Review F09 O’Brien
FM Lial 9th
x
P(x)
4
.4
6
.4
8
.05
10
.15
19.
Find the expected value for the random variable.
20.
Find the expected value for the random variable x having the probability function shown in the graph.
21.
A die is biased in such a way that the probability of a particular side turning up is proportional to the number of pips
on that side. Such a die is to be rolled and the number of pips on the top side recorded.
a.
Construct the probability distribution and the corresponding histogram for this experiment.
b.
What is the expected number of pips?
c.
What are the odds for the number of pips being at least 4?
22.
A raffle offers a first prize of $1,000, two second prizes of $300 each, and 20 third prizes of $10 each. If 10,000
tickets are sold at 50¢ each, find the expected payback for a person buying one ticket. Is this a fair game?
9.1
Frequency Distributions; Measures of Central Tendency
23a.
Given the adjacent set of data, group the data into six intervals
starting with 0 – 24; prepare a frequency distribution with
columns for intervals and frequencies; construct a frequency
histogram; and construct a frequency polygon.
What is the mean for the original data?
8
26
41
123
3
87
103
46
74
14
99
148
115
127
65
44
86
35
72
81
143
39
95
84
126
57
79
83
91
14
24
135
76
82
39
100
What is the mean for the grouped data?
24.
Find the mean, median, and mode of the data and decide which best describes the data. Explain your reasoning.
Salaries of $30000, $60000, $65000, $63000, $80000
25.
Find the mean, median, and mode of the data and decide which best describes the data. Explain your reasoning.
Test scores: 76, 83, 92, 75, 83, 98, 65, 73, 82, 56, 43, 78, 80
26.
A random sample of 28 households was questioned about the number of can openers each household possessed.
Their responses are shown in the adjacent table.
a. Prepare a frequency distribution for these data.
2
2
2
4
2
3
0
b. Construct a frequency histogram for the data.
1
3
1
1
1
3
3
c. What percentage of homes have exactly two can openers?
3
1
2
2
1
2
2
d. What percentage of homes have at least two can openers?
2
3
3
1
2
2
1
e. What are the odds that a home had three or more can openers?
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Final Exam Review F09 O’Brien
FM Lial 9th
9.2
Measures of Variation
On problems 27 and 28, do the following.
a. Calculate the appropriate mean, x or μ .
b. Calculate the appropriate deviation from the mean, x  x or x  μ .
c. Find the sum of the deviations from the mean,  x  x  or  x  μ  .
d. Find the squares of the deviations from the mean, x  x 2 or x  μ 2 .
e. Find the sum of the squares of the deviations from the mean,  x  x 2 or  x  μ 2 .
f. Find the appropriate variance (sample or population).
g. Find the appropriate standard deviation (sample or population).
h. Find the range of the data set.
27.
Sample: 12, 18, 20, 30, 32, 33, 80, 82, 84, 86
28.
Population: 50, 51, 52, 52.5, 53, 53.5, 54, 54.5, 55, 55.5
29.
Find the mean and standard deviation for the adjacent grouped
data by expanding the table to include columns for x, xf, x 2, and
fx2.
30.
A machine produces what is called a “3-inch” bolt. From a day’s production run, five bolts are chosen at random and
measured for length. The results, in inches, are 3.01, 2.97, 3.10, 3.11, 2.98.
a.
Find the mean and standard deviation of these lengths.
b.
How many lengths are within one standard deviation of the mean?
9.3
The Normal Distribution
31.
Find the percent of the area under a normal curve between the mean and the given number of standard deviations
from the mean.
a.
32.
–2.57
z = –.35 and z = 1.67
b.
z = –2.31 and z = –.56
c.
z = 1.28 and z = 3.05
Find a z-score satisfying the following conditions.
a.
34.
b.
Frequency
5
7
4
11
4
5
Find the percent of the total area under the standard normal curve between each pair of z-scores.
a.
33.
1.35
Interval
0 – 24
25 – 49
50 – 74
75 – 99
100 – 124
125 – 149
11.9% of the total area is to the left of z
b.
16.6% of the total area is to the right of z
A government agency checked the weights of bags of peanuts labeled “net weight 14 oz.” The agency found that the
weights on the bags that were checked were normally distributed, with a mean of 14.1 oz. and a standard deviation of
.2 oz. Based on this information, what percentage of the bags of these peanuts:
a. will weigh at least 14 oz.?
b.
will weigh between 13.8 and 14.5 oz.?
c. will weigh more than 14.3 oz?
d.
will weigh between 14.4 and 14.7 oz?
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Final Exam Review F09 O’Brien
FM Lial 9th
9.4
Normal Approximation to the Binomial Distribution
35.
Suppose 12 coins are tossed. Find the probability of getting the following results first using the binomial probability
formula and then using the normal curve approximation.
a.
exactly 5 heads
b.
more than 9 tails
c.
fewer than 4 heads
36.
A single die is thrown 144 times. Use the normal curve approximation to find the probability of getting more
than 37 ones.
37.
A new drug cures 73% of the patients to whom it is administered. It is given to 20 patients. Find the probabilities that
among these patients, the following results occur.
a.
all are cured
b.
none are cured
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Final Exam Review F09 O’Brien
FM Lial 9th
Answers
1.
300
2.
a.
3.
1728
4.
a.
5.
840
6.
336
b.
1,816,214,400
35,152
b.
27,600
c.
1104
a.
3276
b.
351
c.
3682
7.
a.
9
8a.
permutations; 720
9.
130,945,815
10.
2
5
11.
a.
12.
6
 .0014
4165
13.
a.
14.
.7063
15.
a.
16.
.925
17.
.826
b.
7
 .318
22
364
 .376
969
1
 .000129
7776
b.
combinations; 4845
c.
6
combinations; 792
d.
b.
5
 .152
33
c.
28
 .848
33
b.
2834
 .975
2907
c.
793
 .818
969
b.
3125
 .402
7776
c.
625
 .161
3888
permutations; 5040
d.
763
 .196
3888
18.
x
0
1
2
3
19.
5.9
20.
2.5
P(x)
1  .0333
30
3  .3
10
1  .5
2
1  .1667
6
P(x ≤ 2) =
5
6
 .8333
E(# of reds) = 1.8
6
Final Exam Review F09 O’Brien
FM Lial 9th
21.
x
1
2
3
P(x)
1  .04762
21
2  .09524
21
3  1
21 7
 .14286
4
5
4  .19048
21
5  .23810
21
O(x ≥ 4) = 5 to 2
22.
–32¢; No, this is not a fair game.
23.
Interval
0 – 24
25 – 49
50 – 74
75 – 99
100 – 124
125 – 149
E(x) =
Interval
0 – 24
25 – 49
50 – 74
75 – 99
100 – 124
125 – 149
3, 8, 14, 14, 24
26, 35, 39, 39, 41, 44, 46
57, 65, 72, 74,
76, 79, 81, 82, 83, 84, 86, 87, 91, 95, 99
100, 103, 115, 123
126, 127, 135, 143, 148
6
6  2
21 7
 .28571
13
 4.333
3
Frequency
5
7
4
11
4
5
Mean for original data
Mean for grouped data
74
73.806
24.
mean: $59,600
best measure: median
median: $63,000
mode: none
reason: outliers might be skewing the mean
25.
mean: 75.7
best measure: mean
median: 78
mode: 83
reason: the mean is very close to the average of the high score and low score
26a.
b.
c.
39.3%
d.
.679
e.
8 to 20
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Final Exam Review F09 O’Brien
FM Lial 9th
27.
f.
g.
h.
b – e.
x = 47.7
8704.1
s 
 967.122
9
a.
2
s  967.122  31.09859
range = 74
x– μ
x
28.
g.
h.
a.
μ  53.1
f.
σ2 
b – e.
27 .9
 2.79
10
σ  2.79  1.6703
range = 5.5
–3.1
–2.1
–1.1
–.6
–.1
.4
.9
1.4
1.9
2.4
50
51
52
52.5
53
53.5
54
54.5
55
55.5
531
0
x μ 2
9.61
4.41
1.21
.36
.01
.16
.81
1.96
3.61
5.76
27.9
29.
Interval
0 – 24
25 – 49
50 – 74
75 – 99
100 – 124
125 – 149
Frequency
5
7
4
11
4
5
36
x
12
37
62
87
112
137
xf
60
259
248
957
448
685
2657
x2
144
1369
3844
7569
12,544
18,769
fx2
720
9583
15,376
83,259
50,176
93,845
252,459
mean of grouped data  73.8
standard deviation of grouped data  40.1
30.
a. x  3.034 ; s = .066558
b. The range of lengths within one s.d. of the mean is 2.967 up to 3.10. Four of the five bolts fall within this range.
31
a. 41.15%
b.
49.49%
32.
a.
58.93%
b.
27.73%
33.
a.
z = –1.18
b.
z = .97
34
a. 69.15%
b.
91.04%
35.
a.
P(5H) =
b.
P(> 9T) = P(10, 11, 12) =
c.
P(< 4H) = P(0, 1, 2, 3) =
99
512
c.
9.92%
c.
15.87%
d.
6.55%
 .1934; P(4.5 < x < 5.5) = P(–.87 < z < –.29) = .3859 – .1922 = .1937
79
 .0193; P(x > 9.5) = P(z > 2.02) = 1 – P(z
4096
299
 .0730; P(x < 3.5) = P(z < –1.44) = .0749
2048
36.
P(x > 37.5) = P (z > 3.02) = 1 – P(z ≤ 3.02) = .0013
37.
a.
P(19.5 < x < 20.5) = P(2.47 < z < 2.97) = .0053
b.
P(x = 0) = 4.239 x 10–12  0
≤ 2.02) = .0217
8