MDM 4U
Culminating Tasks
Mathematics of Data Management
End of Year Review
This review will help you prepare for the final exam. In answering these questions you may consult with
others but it is essential that you know how to do these questions on your own before the exam.
Part 1 - Combinatorics
1.
Given the ten digits 0, 1, 2, … ,9, find how many 4-digit numbers can be made from them with each
of the following restrictions:
a)
No digit may be repeated.
b)
Repetitions are allowed.
c)
No digit is repeated and the number is odd.
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_________________
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2.
Given five flags of different colours, how many signals can be made if at least three flags must be
used for each signal. A “signal” is a particular arrangement of flags on a vertical pole.
_________________
3.
A student has six examinations to write and there are ten examination periods available. How many
different examination timetables can he get?
4.
In how many ways can eight books be arranged on a shelf if two specified books must not be side by
side?
_________________
5.
In how many ways can a selection of fruit be made from six plums, five apples and eight oranges?
Include the possibility that not all types of fruit are selected; also, selecting no fruit at all is not an
option.
_________________
MDM 4U
Culminating Tasks
6.
Given a set of 12 points, no 3 of which are on a straight line, find the number of triangles that can be
formed using the points as vertices.
7.
How many selections of one or more letters can be made from 3 A’s, 7 B’s and/or 11 C’s?
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8.
Find the number of ways in which 16 different objects can be divided into 3 parcels containing 4, 5
and 7 objects.
9.
How many different characters can be transmitted by Morse code if each character is represented by
not more than 5 dots and/or dashes?
10.
How many ways can the letters of the word HOPEFUL be arranged if the consonants must remain in
the original order?
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11.
Four co-workers are to be selected from a pool of 10 workers to attend a conference. However, two of
the co-workers cannot both be selected at the same time. Given this restriction, in how many ways
can the co-workers be selected?
12.
In how many ways can eight identical, delicious, fried tofu squares be distributed among three
people, including the possibility that a person gets no squares?
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13.
How many ways can the letters of the word IMPONDERABLY be arranged to that any arrangement
does not include the words POND or ABLE?
MDM 4U
Part 2 – Probability and Probability Distributions
14.
Culminating Tasks
________________
What is the probability of winning the game rock-paper-scissors:
a) On the first go?
b) On the second go (assuming the first go was a tie)?
_______________
15.
A tetrahedral die has four faces each numbered 1 through 4. What is the probability of getting a sum
of 5 on a single toss of two tetrahedral dice?
________________
16.
A bicycle shop has ten inner tubes for sale but one of them has a leak. What is the probability that the
fourth person to buy an inner tube gets the one that is leaky?
_______________
17.
Two factories, I and II, produce 30% and 70% of the daily outputs of brake calipers for a particular
company. Factory I produces 5% defectives and Factory II produces 8% defectives. A brake caliper is
sampled from the day’s production. What is the probability that it is from Factory II and defective?
18.
An e-mail is sent to two people. The probability that it will be read before the weekend by at most
one of the two people is 0.7 and the probability that it will be read by at least one of the two people is
0.5. What is the probability that it will be read before the weekend by the two people?
________________
19.
Art, Becky, Chris, Deidre and Engel go out for dinner each Thursday. Each time, they randomly pick
two of the group to pay for the dinner. What is the probability that Chris or Deirdre (or both) pay for
this Thursday’s dinner?
________________
MDM 4U
20.
Culminating Tasks
In a large apartment complex, 40% of the residents read The Toronto Sun and 25% read The Toronto
Star. Five percent of the residents read both papers.
a)
Find the probability that a randomly chosen person from this complex reads exactly one of the
two papers.
________________
b)
Given that one of the readers of The Toronto Sun has been randomly selected, what is the
probability that this person reads The Toronto Star?
________________
21.
At a large university, 49% of the students are female, 69% live in dorms, and 42% are from the
province of Ontario. Further, 34% of the students are females who live in the dorms, 19% are females
who are from the province of Ontario, and 16% are from the province of Ontario who live in the
dorms. Finally, 7% of the students are females who live in the dorms and are from the province of
Ontario. What is the probability that a randomly chosen student from this university is a male who
does not live in the dorms and is not from the province of Ontario?
22.
A box contains 100 gaskets but 15 of them are defective. You purchase 8 of the gaskets which are
selected at random from the box. What is the probability that exactly 2 of them are defective?
________________
23.
In Canada, roughly 75 percent of adults wear glasses or contact lenses. A random sample of 10 adults
in the Canada will be selected. What is the probability that fewer than 8 of the selected adults wear
glasses or contact lenses?
________________
24.
In five-card poker, five cards are dealt from a standard deck of 52 cards. Which is more probable: a
full house (three-of-a-kind and two of a kind) or four-of-a-kind? Do the calculations to justify your
answer.
________________
MDM 4U
25.
Culminating Tasks
Airlines routinely overbook flights because they expect a certain number of no-shows. An airline runs
a 5 P.M. commuter flight from Washington, D.C., to New York City on a plane that holds 38
passengers. Past experience has shown that if 41 tickets are sold for the flight, then the probability
distribution for the number who actually show up for the flight is as shown in the table below. Let X
be the number of people who actually show up to the flight and assume 41 tickets are sold for each
flight.
36
37
38
39
40
41
X
P( X )
0.46
0.30
0.16
0.05
0.02
0.01
a) There are 38 seats on a flight. What is the probability that all passengers who show up for
the 5 P.M. flight will get a seat?
b) What is the expected number of people to show up for a flight?
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26.
The nine delegates to the United States-Mexico-Canada Agreement (USMCA) talks
included 2 officials from Mexico, 3 officials from Canada, and 4 officials from the United States.
During the opening session, three of the delegates fall asleep. Assuming that the three sleepers were
determined randomly, what is the probability that exactly two of the sleepers are from the same
country?
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Part 3 – Statistics and Probability
27.
Judith scored 680 on the SAT math test and 27 on the ACT math test. Scores on the math section of
the SAT vary from 200 to 800 with a mean of 514 and a standard deviation of 117. Scores on the
math section of the ACT vary from 1 to 36, with a mean of 21 and a standard deviation of 5.3.
Calculate Judith’s standardized score on each test (z-score) and determine which of her test scores
was better.
________________
28.
Jennifer and her son Leo graduated from the same high school. Jennifer’s GPA was 3.9 and Leo’s
was 4.2. However, when Jennifer graduated, the mean GPA was 2.8 with a standard deviation of 0.6.
When Leo graduated, the mean GPA was 3.2 with a standard deviation of 0.7. Based on this
information, which person had a higher score relative to their peers? Justify your answer.
________________
MDM 4U
29.
Culminating Tasks
Suppose a person’s diastolic blood pressure X (in millimetres of mercury) can be modelled by a
normal distribution, X ~ N(70, 400).
a)
A healthy diastolic pressure for an adult is less than 80. Determine P(X < 80) for a randomly
chosen person.
b)
A diastolic pressure between 80 and 90 indicates borderline high blood pressure. What percent
of adults have borderline high blood pressure?
________________
30.
In the United States, egg sizes are set by the Department of Agriculture. A “large” egg, for example,
weighs between 57 and 64 grams. Suppose the weights of eggs produced by hens of a particular
farmer are normally distributed with a mean of 55.8 grams and a standard deviation of 7.5 grams. For
a randomly selected egg from this farmer’s hens, what is the probability that it is a “large” egg?
________________
31.
32.
The army reports that the distribution of head circumference among soldiers is approximately normal
with mean 57.912 cm and a standard deviation of 2.794 cm. Helmets are mass produced for all except
the smallest 5% and the largest 5% of head sizes. What range of head sizes get custom-made helmets?
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Scores on the Wechsler Adult Intelligence Scale (a standard IQ test) for the 20-to-34 age group are
normally distributed, X ~ N(110, 625). MENSA only admits members who score in the top 2% on IQ
tests. What score on the Wechsler Adult Intelligence Scale would an individual aged 20 to 34 have to
earn to qualify for MENSA membership?
________________
33.
Suppose 11% of the students at UTS are left-handed. In a random sample of 100 students, what is the
probability that at least 20 of the students are left-handed. (In answering, use a normal approximation
to a binomial.)
________________
MDM 4U
Culminating Tasks
34.
A multiple-choice test consisting of 40 questions is given to a Data Management class. Each question
has four choices and there is no penalty for guessing. What is the probability of passing the test by
random guessing? (In answering, use a normal approximation to a binomial.)
35.
In the United States, 20% of adults ages 25 and older have never been married (more than double the
figure recorded for 1960). In a random sample of 50 U.S. adults ages 25 and older, what is the
probability that at most 5 have never been married?
Part 4 – One-Variable and Two-Variable Statistics
36.
Suppose you are writing a multiple-choice test with 20 questions, each having a choice of 5 answers.
Now suppose that you make random guesses on the test. Intuitively, you would assume that your
mark will be somewhere around 4 out of 20 since there is a 1 in 5 chance of guessing right on each
question. However, it is possible that you could get any number of the questions right—anywhere
from zero to a perfect score. A simulation was designed to model the guesses made on the test. The
results of 100 simulations are shown.
a) Describe the distribution above.
b) Based on this simulation, estimate the probability of getting less than five answers correct.
MDM 4U
37.
Culminating Tasks
The table and scatter plot show data for the full-time employees of a small company.
a) The linear correlation coefficient for this data set is r = 0.982. Interpret this value.
b) The equation of the LSRL is given by (income) =
−0.85 + 1.15(age) .
Interpret the meaning of the slope in this context.
c) Use the LSRL to predict the income for a new employee who is 21 and an employee retiring
at age 65.
d) Calculate the residual for the 33-year-old employee.
38.
Match each of the following coefficients of determination with one of the diagrams below.
a) 0
b) 0.5
c) 0.9
d) 1
MDM 4U
Culminating Tasks
Answers
Part 1 – Combinatorics
1. a) 4536
b) 9000
c) 2240
2. 300
3. 151 200
4. 30 240
5. 377
6. 220
7. 383
8. 1 441 440
9. 62
10. 210
11. 182
12. 45
13. 478 276 560
Part 2 – Probability and Probability Distributions
14. a) 1/3
b) 1/9
15. 1/4
16. 1/10
17. 7/125
18. 0.3
19. 7/10
20. a) 0.55
b) 1/8 or 0.125
21. 2% or 0.02
22. 0.247
23. 0.47 (Use Binomial Theorem and Compliments: 1 – [P(X = 8) + P(X = 9) + P(X = 10] )
24. Full House
25. a) 0.92
b) ~36.9 passengers
26. 55/84
Part 3 – Statistics and Probability
27. SAT
28. Jennifer
29. a) 0.691
b) 0.150
30. 0.299
31. <53.316 or >62.508 (Note: standard deviation is meant to be in cm)
32. 161.344
33. 0.0033
34. 0.00026
35. 0.049 (discrete binomial calculation); 0.056 (normal approximation to the binomial)
Part 4 – One-Variable and Two-Variable Statistics
36. a) Skewed Right, no obvious outliers, range of 10 b) ~60%
37. a) Strong, positive, linear correlation
b) For every year increase in age, our model predicts an average salary increase of $1150.
c) 21-year-old = $23,300 and 65-year-old = $73, 900.
d) – 4.1. Roughly $4100 below predicted value.
38. a) iii b) i c) iv d) ii