Unit 1 – Counting and Probability
1.
Determine the probability of drawing a face card and then an ace from a regular deck of cards if the face card is
not returned.
2.
In a hat, there are 2 nickels and 2 dimes. If two coins are chosen at random at the same time, what is the
probability that both are nickels?
3.
An artist wins a prize 60% of the time when she enters an art show. Draw a tree diagram and use it to determine
the probability that she will win exactly once out of two art shows.
4.
A bag of marbles contains 4 blue marbles, 7 red marbles, and 3 yellow marbles. A marble is drawn and then
replaced. A second marble is then drawn. Determine the probability of drawing a red marble on the first draw,
and a yellow marble on the second draw.
5.
A prize bag has 5 CDs, 7 video games, and 4 DVD movies. Each winner picks one prize from the bag without
looking in the bag. Determine the probability of the first two winners both choosing DVD movies as prizes.
6.
In a high school of 1000 students, 355 are playing sports and 233 are in a school band. If there are 175 students
playing sports and are in a school band, how many are in neither activity?
7.
Let A be the set of even numbers between 1 and 20. Let B be the set of all numbers evenly divisible by three
between 1 and 20. Find n( A  B ) .
8.
A survey of a machine shop reveals the following information about its employees:
44 employees can run a lathe
49 employees can run the milling machine
56 employees can operate a punch press
27 employees can run a lathe and a milling
machine
19 employees can run a milling machine and
operate a punch press
24 employees can run a lathe and operate a
punch press
10 employees can operate all three machines
9 employees cannot operate any of the three
S
a. Organize the given information in the Venn Diagram above
b. How many people are employed the machine shop?
c. How many run only the lathe?
9.
An advertiser is told that 70% of all adults in the Greater Toronto Area (GTA) read the Toronto Star and 60%
watch CityTV. She is also told that 40% do both. If she places an advertisement in the Toronto Star and runs a
commercial on CityTV, what is the probability that a person selected at random in the GTA will see at least one
of these?
10. What is the probability of drawing two queens in a row from a standard deck of 52 cards, if the first card is not
replaced?
11. A die is rolled and a card is drawn randomly from a regular deck of cards at the same time. Determine the
probability of rolling an even number and drawing a black ace.
12. According to the following Venn diagram, what is the value of
a.
n( A  B)
c. n A  D
b. P B  C'
d. P B  C  D
e. shade in the region defined by ( A  B )  D '
13. At a school of 1400 students, 800 are currently enrolled in English, 740 are
enrolled in Mathematics, and 325 are enrolled in both subjects. How many
students are enrolled in English or Mathematics?
14. Customers at a restaurant are surveyed to determine if they drink pop and eat ice cream. The probability of a person drinking
pop and eating ice cream is determined to be 0.2. The probability of a person eating ice cream given if they drink pop is 0.4.
Determine the probability of a person drinking pop.
15. A survey is done and results are tallied below.
Hair Colour
Child
Blond Hair
23
Brown Hair
16
Other
24
Teenager
34
15
21
Adult
67
83
70
If a person is selected at random, determine the probability that
a) they have blond hair given that they are an adult
b) they have blond hair
c) they have brown hair and are an adult
16. A group of 200 candidates apply for a job. Only 10 will be given interviews and only 3 of those candidates will be given
second interviews. If the selection process is completely random, what is the probability of being given a second interview?
17. Out of a group of 100 people, 63 are adults and 37 are children. Of the adults, 31 have brown eyes and of the children, 29
have brown eyes. If a person is selected at random, determine the probability of that person being a child if the colour of eyes
is known to be brown.
18. A picnic cooler contains different types of cola: 14 regular, 5 cherry, 11 diet, 4 diet cherry, 8 caffeine-free, and some
caffeine-free diet. You pick a can of cola without looking at its type. There is a 45% chance that the drink selected is not
diet. How many caffeine-free diet colas are in the cooler?
19. A box contains a group of 24 blocks. Some are plain red, some are plain yellow, and some are a mixture of the two colours.
1
1
The probability of drawing a plain red block is . The probability of drawing a red and a yellow block is
. Determine
12
3
the number of plain yellow blocks.
20. A game consists of tossing a coin, cutting a deck of cards, and rolling a die. Bonus points are given if you repeat exactly the
outcomes of your previous turn. Determine the probability on your first turn that you have tails, an ace, and an even number
on the die, and then you receive bonus points in the next two turns.
Unit 2 – Combinations and Permutations
1. Determine the number of arrangements for the word BOXCAR.
2. Determine the number of arrangements for the word CANADA.
3. Determine the number of ways that a prime minister, secretary, treasurer, and publicity minister could be selected
from an art club of 12 members.
4. Determine the number of ways the 8 members of the Junior Jazz Band can stand in a line if Val must be first, Tim
sixth, and Tricia last.
5. Evaluate each expression. Show your steps.
a) P(5, 3)
6
 3
b) 5 
c) 3! – 2 x 4! + P(7, 0) – C(3, 3) +0!
6. Write out the first six rows of Pascal’s triangle.
7. What is the sum of the entries in the seventh row of Pascal’s triangle?
8. Expand and simplify using combinations to find the coefficients of each term.
a) (a + b)4
b) (x -3)5
9. A committee of students and teachers is being formed to study the issue of student parking privileges. Fifteen staff
members and 18 students have expressed an interest in serving on the committee. In how many different ways
could a five-person committee be formed if it must include at least one student and one teacher.
10. The Greek alphabet contains 24 letters. How many different Greek-letter fraternity names can be formed using
either two or three letters? (Repetitions are allowed.)
11. Tom arrives at the giant auction sale late in the afternoon. There are only five items left to be sold. How many
different purchases could he make?
12. In how many ways can a principal select a graduation committee consisting of two teachers and 6 students if there
are 8 teachers and 10 students who are volunteering for the positions?
13. In how many ways can 10 students be arranged in a line if Rhea must be first and Ava last?
14. A package of 20 transistors contains fifteen that are perfect and five that are defective. In how many ways can five
of these transistors be selected so that at least three are perfect?
15. Marla’s bag of marbles contains two red, three blue and five green marbles. If she reaches in to pick some without
looking, how many different selections might she make?
16. In how many ways can a team of six female volleyball players be chosen to start the game from a roster of 15
players.
17. In how many ways can the six members of the hockey team line up at the blue line so that the two defensemen are
not side by side?
18. Suppose you are designing a coding system for data relayed by a satellite. How many ten-digit codes can you
create if the first three digits must contain the numbers 1, 3 or 6?
19. A pizza place offers 8 different toppings. Determine the number of different pizzas you can order if you had to
choose at least one topping.
20. The game of euchre uses only 9s, 10s, jacks, queens, kings and aces. How many five-card hands have all red
cards?
21. Seven managers and eight sales representatives volunteer to attend a trade show. Their company can afford to
send five people. In how many ways could they be selected if there must be at least one manager and one sales
representative chosen?
22. At the Toyota dealership, eight cars are to be parked side-by-side. In how many ways can this is done if the three
red cars must be parked beside each other?
23. Solve for n.
 n 
  10
 n  2
a) 
 n  1

 3 
b) P(n, 4)  60
c) 120C(n, 3) = P(n, 5)
Unit 3 - Analyzing One Variable Data Sest
1.
A sample of high school students is interviewed to determine the most popular movie star. Is the variable of
interest quantitative or qualitative?
2.
Identify the variable, number of cancer deaths in each province last year, as continuous or
discrete.
3.
A systematic random sample is to be taken of 25 students from a school of 1500 students. The students are labeled
by a student number. What is the appropriate sampling interval?
4.
A survey of people living in a high-rise residence is to be conducted. The residence has 10 floors, 20 rooms per
floor, and 2 people per room. What type of sampling method selects 2 floors and surveys everyone on those
floors?
5.
A sports magazine wants to know about its subscribers’ level of active participation in team sports. It sends a
survey to 1000 randomly selected subscribers. What type of bias is most likely to occur?
6.
A type of question where respondents reply in their own words is called…..?
7.
What shape of distribution best describes the following data?
0
20
Value
Frequency
1
22
2
13
3
8
4
4
5
1
8.
What is the standard deviation (to one decimal place) for the following data set?
13, 27, 31, 34, 38, 48, 59, 77
9.
Find the mean, median, and mode of the pop quiz marks shown below. (3 marks)
0
4
Mark
Frequency
Mean _________
10.
2
3
3
6
4
7
Median _________
5
1
Mode __________
The following chart lists the ages of students on the student council. Calculate the mean and the standard
deviation of the ages, to one decimal place. Show your work.
14
3
Age (years)
Frequency
11.
1
2
15
6
16
10
17
15
The daily maximum temperatures (in °C) from the past 14 days are listed below.
Find Q1, Q2, and Q3 and use them to find the interquartile range.
16, 18, 20, 23, 26, 19, 15, 14, 13, 15, 15, 11, 7, 9
12.
The speeds of 24 motorists ticketed for exceeding a 50-km/h limit are listed below.
71
88
66
75
68
72
82
80
74
71
70
93
78
82
80
73
91
65
75
90
69
74
74
79
Construct a histogram using six uniform intervals. Organize the data first into a frequency table.
13.
In a certain course, a student’s final mark is computed using the following weights: Quizzes 10%, Tests 40%, and
Examination 50%. When Alfredo took this course last semester, he had a mark of 81% on his quizzes, 75% on
his tests, and 70% on his final exam. Find Alfredo’s final mark in the course. Round to the nearest whole
number.
14.
A survey is designed to determine how Canadians will vote in the next election. It surveys 100 randomly selected
adults from each province and territory. Is this likely to give biased results? If it is, what type of bias will exist?
15.
What is the best measure of central tendency for a data set containing qualitative data? Explain.
16.
List 2 characteristics of good survey questions and 2 things that good surveys avoid.
17.
List and describe the different types of bias.
18.
Define and provide examples of discrete, continuous, quantitative and qualitative variables.
19.
A box contains 10 balls each labeled with a different number from 1 to 10. Three different balls are selected at
random and the largest of the 3 numbers is recorded. The balls are returned and the process is repeated a large
number of times. A frequency histogram for the numbers recorded is constructed. What shape would you expect it
to have? Explain.
20.
Explain how you would create a box and whisker plot
21.
List and describe the different types of sampling
Unit 4 - Analyzing Two Variable Data Sets
1. Give examples of the following correlations:
a. strong, positive
b. strong, negative
c. weak, positive
d. weak, negative
2. A researcher is curious to find out what effect classical music has on people’s level of relaxation (as measured by
heart rate). He suspects that listening to classical music will make people feel more calm and relaxed. He lets
one group listen to classical music for one hour. He lets another group sit in a quiet room for one hour (i.e they
hear no music). After one hour, he monitors the heart rate of each participant to measure their level of relaxation.
Identify:
a) the experimental group
b) the control group
c) the independent variable
d) the dependent variable
3. Classify the relationships in the following situations as either, cause-and-effect relationship, presumedrelationship, common-cause factor, or accidental relationship.
a) A students math mark changes based on amount of time spent studying.
b) The price of lettuce and housing prices have a strong positive correlation over many years.
c) Job performance has a positive correlation with academic achievement.
d) Sales of dishwashers had a negative correlation with level of physical fitness
4. Classify the relationships in the following situations as either, cause-and-effect relationship, presumedrelationship, common-cause factor, or accidental relationship. Use each answer only once.
a) A student’s math mark has a positive correlation with their science mark.
b) The profit of oil companies has positive correlation with the price of gas.
c) The size of the corn harvest has a positive correlation with the size of the apple harvest.
d) Sales of toasters have a strong negative correlation with the number of points scored at a football game.
5. At 1821 feet tall, the CN Tower in Toronto, Ontario, is the world’s tallest self-supporting structure. Suppose you
are standing in the observation deck on top of the tower and you drop a penny from there and watch it fall to the
ground. The table below shows the penny’s distance from the ground after various periods of time (in seconds)
have passed.
Time
Distance
a. Create a scatter plot of the data shown in the table.
(seconds)
(feet)
b. Describe any patterns and correlations in the data.
0
1821
c. Draw the median-median line.
2
1757
d. Use your graph to predict the height after 5 seconds.
4
1565
6
1245
8
797
10
221
Unit 5 – Discrete Probability Distributions
1. The manager of a telemarketing firm conducted a time study to analyze the length of time employees spent engaged in a
typical sales-related phone call. The results were as follows:
Time (min)
Frequency
1
15
2
14
3
18
4
20
5
13
6
10
(a) Define the random variable
(b) Find the probability distribution for this data set
(c) Determine the expected value
2. The faces of a 12-sided die are numbered from 1 to 12. What is the probability of rolling 9 at least twice in ten
tries?
3. A carton contains 24 light bulbs, 3 of which are defective. What is the probability that, if a sample of 6 is chosen
at random from the carton of bulbs, 1 will be defective?
4. A factory making circuit boards has a defect rate of 2% on one of its production lines. An inspector tests
randomly selected circuit boards from this production line.
(a) What is the probability that the first defective circuit board will be the sixth one tested?
(b) What is the expected waiting time until the first defective circuit board?
5. Ten percent of a country’s population are left-handed.
(a) What is the probability that 5 people in a group of 20 are left handed?
(b) What is the expected number of left handed people in a group of 20?
6. 1000 fish in one of the lakes of Ontario were caught and tagged by the Department of Fisheries. After a while a
new catch of 500 fish was caught and it was found that 60 among them had tags. Estimate the number of fish in
the lake.
7. At the Statsville County Fair, there is a 20% chance of winning a prize in the ring-toss game. If someone plays
the ring toss game 20 times, what is the probability that they will win a prize exactly 5 times?
8. A rectangular area is to be enclosed with 12 m of fencing. The length of the rectangle is to be an integer length
and is to be chosen randomly. What is the expected area of the rectangle?
9. A restaurant knows that 18% of parties making reservations do NOT show up. One evening, there were 30
reservations. Find the probability that everyone showed up.
10. A committee of four people is to be chosen randomly from four males and six females. Identify which type of
distribution you would use to find the probability distribution for the number of females on the committee, and
explain your choice.
11. A coin is tossed 20 times. Identify which type of distribution you would use to find the probability distribution
for the number of tails tossed, and explain your choice.
Unit 6 – Continuous Probability Distributions
1. A children’s shoe maker knows that the shoe sizes of children under 10 are normally distributed with a mean of 5
and a standard deviation of 1.5. To guarantee that 99.7% of children can buy a pair of his shoes that are not too
small, what sizes must be manufactured?
2. Batting averages in a softball league are normally distributed with a mean of 0.260 and a standard deviation of
0.033. What percent of players have a batting average less than 0.300?
3. The average time it takes to speak to a customer service representative of a credit card company is 4 minutes.
Calculate the probability of talking to a customer service representative in less than 2 minutes.
4. The time that a certain top sprinter takes to run the 100-m dash is normally distributed with a mean of 9.8 s and a
standard deviation of 0.2 s. In what percent of his sprints will his time be less than 10.0 s?
5. A study of 55 patients with lower-back pain reported that the mean duration of the pain was 17.6 months, with a
standard deviation of 5.1 months. Assuming that the duration of this problem is normally distributed in the
population, determine a 99% confidence interval for the mean duration of lower-back pain in the population.
6. A study found that the average time it took for a university graduate to find a job was 5.4 months, with a standard
deviation of 0.8 months. If a sample of 72 graduates were surveyed, determine a 99% confidence interval for the
mean time to find a job.
7. IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. The Regular People Club
does not accept people with IQs lower than 85 or higher than 115. How many people in a group of 400 would you
expect to be refused membership in the club?
8. A lake in Ontario has a mean June water temperature of 20.3°C and a standard deviation of 2.1°C. How many
days in June (30 days) would you expect to find comfortable for swimming if you will only swim when the water
temperature is between 19°C and 25°C?
9. A single die is rolled 30 times. Find the probability of getting a 2 more than 10 times. Show all steps.
10. A store manager believes that 42% of her customers are repeat business. What is the probability that out of the
next 500 customers, more than 200 are repeat businesses? Show all steps.
11. A traffic study showed that vehicle speeds on a particular highway were normally distributed with a mean of
102.5 km/h and a standard deviation of 5.1 km/h. What percent of vehicles had a speed between 92.3 km/h and
107.6 km/h?
12. The masses of Burgerville’s burgers are normally distributed. Of these burgers, 33% have masses greater then
253.52 g. If the average mass of the burgers is 250 g, find the standard deviation.
13. An industrial-safety inspector wishes to estimate the average noise level, in decibels (dB), on a factory floor. She
knows that the standard deviation is 8 dB. She wants to be 90% confident of an accuracy of  2 dB . How many
noise-level measurements should she take?
14. For X~N( x ,  2 ), what percent of the data fall between x  2 and x   ? Include a diagram in your
explanation.
Exam Review Answers
Unit 1
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
Unit 3
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
13.
14.
15.
19.
0.018
0.167
0.48
3/28
1/20
587
13
b. 89 c. 3
0.9
1/221
1/52
a. 28 b. 10/43 c. 0 d. 2/43
1215
1/2
a. 67/220 b. 124/353 c. 83/353
3/200
0.483
18
14
1/140 608
Qualitative
Discrete
60
Cluster
Sampling
Open
Right-skewed
18.7
Mean=2.6, median=3, mode=4
Unit 2
1.
2.
3.
4.
5.
6.
7.
8.
720
120
11 880
120
a. 60 b. 100 c. -41
1
11
121
1331
14641
1 5 10 10 5 1
1 6 15 20 15 6 1
128
a) a 4  4a3b  6a 2b2  4ab3  b 4
b) x 2  15 x 4  90 x3  270 x 2  450 x  243
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
Unit 4
1.
2.
3.
x  16.1,   1.0
4.
Q1=12.5, Q2=15, Q3=19.25,
IQR=6.75
73%
Household
Mode
Left-skewed
5.
225 765
14 440
31
5880
40 320
14 378
71
5005
480
60 000 000
255
792
2926
4320
a. 5 b. 10 c. 8
28
5039
10 toppings
Answers vary
a. group with music
b. group without music
c. presence of classical music
d. level of relaxation (heart rate)
a. cause-and-effect
b. common-cause
c. presumed-relationship
d. accidental-relationship
a. presumed-relationship
b. cause-and-effect
c. common-cause
d. accidental-relationship
b. strong, positive
d. approx. 1200 feet
Unit 5
1.
a. X represents the length of time on a sales
call
b.
x
1
2
3
4
5
6
P(x) 15 14 18
20 13 10
90
c.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
90
151
45
0.2003
0.4536
0.01849
a. 0.03192 b. 2
8333
0.175
7 m2
0.0025967
Hypergeometric
Binomial
90
90
90
90
Unit 6
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
13.
14.
15.
0.5-9.5
87%
0.39
84%
15.829    19.371
5.16    5.64
128
22
0.0036
0.829
82%
8
44
81.5%