MDM4U – Final Exam Review
Unit 1: Organized Counting and Permutations
1.
Solve the following without a calculator. Show your work.
13!
110!
a)
b)
9!4!
108!
2.
A Canadian postal code consists of 6 characters of 3 letters alternating with 3
digits. An example of a postal code is M5N 2R6
a) How many possible postal codes can there be if all 10 digits and all 26 letters
can be used? (Note: Digits and letters may be repeated.)
b) How many possible postal codes can be if the digit 0 and the letter O are
excluded and digits or letters may not be repeated?
3.
Five coins are tossed. Give the number of ways the following could occur.
a) 3 heads and 2 tails
b) Exactly one tail
c) At least one tail
4.
Emilio has picked up his textbooks for the seven courses he will study this
year. In how many ways can he arrange them on his bookshelf if he wants to
keep the French and Math texts side by side?
5.
How many different arrangements can be made using all the letters in the
word:
a) SECOND
b) IMPLICIT
c) MISSISSAUGA
6.
Andrea and Ling are evenly matched tennis players. However, each time Ling
1
loses a game his probability of winning the next game is decreased by
. But
5
1
when he wins, his probability of winning the next game increases by.
10
a) Make a tree diagram for a three game sequence and label the diagram with
the probabilities associated with each branch of the diagram.
b) Find the probability that Ling wins at least two games.
c) Find the probability that Andrea wins exactly two games.
7. a) How many ways are there for the black checker ☻to get to the other side of
the board moving only diagonally? You may jump over the white ☺ pieces.
☺
☺
☺
☻
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MDM4U – Final Exam Review
b) Determine the number of possible routes from A to B if you travel only south
and east.
A
B
8.
Answer the following related to Pascal’s triangle.
a) Display the first 5 rows in Pascal’s triangle.
b) What is the sum of the 5th row in Pascal’s triangle?
Unit 2: Combinations
9.
Data obtained from a survey of families were reported as follows:
114 eat meat at least once every day
100 eat bread at least every day
70 eat fruit at least once every day
48 eat meat and bread once a day
41 eat meat and fruit once every day
27 eat bread and fruit once every day
17 eat all three of these foods once a day
a) Show this information on a Venn diagram.
b) What is the minimum number of families surveyed?
c) How many of these families were vegetarians?
10. Given the set {a, b, c, d} how many subsets contain:
a) 0 elements? List the subset(s).
b) 1 element? List the subset(s).
c) 2 elements? List the subset(s).
d) at least 3 elements? List the subset(s).
e) at least 1 element? (use the formula __________)
11. If there are 26 cards in a deck, how many different 5 card bundles could be
made?
12. There are 5 different Math books and 10 different English books. You are to
select 2 Math and 3 English books. How many selections do you have? Show all
of your work and do not use a calculator to answer this question.
13. What is the difference between a permutation and a combination? Give an
example of when you would use each.
14. Ten girls and twelve boys have auditioned for a rock group to be made up of
three boys and three girls. A photographer has made prior arrangements to take
group shots of all the possible winners. How many photos did he take (order
counts)? Hint: He was very busy taking photos for a long time.
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MDM4U – Final Exam Review
15. In how many ways can a committee of five be formed from six men and seven
women if
a) There are no restrictions?
b) The committee must contain exactly three men?
c) The committee must contain at least one man?
d) The men must have a majority?
e) The committee can have only one gender?
16. Rewrite n-1Cn-3 using factorials and then simplify the expression.
17. Determine if the following are mutually exclusive, non-mutually exclusive,
independent or non-independent events. Explain each.
a) Getting a 40% discount when scratching the saving card on a “Bay Day” at the
mall.
b) Katie is really hoping that her parents decide to host her next birthday part at
either the local swimming pool or arcades.
c) A coin is flipped and a die is rolled. Consider the probability of you flipping
heads and rolling a 5 in a single trial.
d) A card is selected from a deck of cards. Consider the probability that either a
diamond or a face card is selected.
Unit 3: Introduction to Probability
18. Consider the problem of 10 people running for an election. Four positions need
to be filled. What is the probability that Mary and Bill are elected President and
Vice-President?
19. Assuming that the probability of giving birth to a boy or a girl is equally likely.
a) Draw a tree diagram to represent the sex of the children of a family with three
children.
b) If a family is selected at random form families with three children, find the
probability that:
i) the children are all the same sex
ii) the oldest is a girl
iii) the family is all boys
iv) there are more boys than girls in the family
v) there is a girl in the family
vi) the middle child is a boy
20. Two dice are rolled.
a) What is the probability of getting a sum of 7?
b) What is the probability of getting a sum of 7 if we know that a 4 has been rolled
on one of the die.
21. There are 10 red balls, 20 black balls and 70 green balls in a jar. Three balls
are being removed without replacement.
a) What is the probability that all of the 3 balls removed are red?
b) Which is more likely, all balls being the same colour or all balls being a
different colour?
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MDM4U – Final Exam Review
Unit 4: Probability Distribution
22. A spinner is divided into 20 equal sectors. The regions are labeled as follows:
6 with $1
4 with $2
3 with $5
2 with $10
5 with $20
a) What is the probability distribution of winnings? Show this in a table.
b) Graph the probability distribution.
c) Find the expected winnings.
d) If Karen charges $5 to play the game, is it a fair game? Explain.
23. A certain surgical operation on the knee is known to be successful 95% of the
time. Seventeen people have had the operation. Find the probability that all
seventeen people have had a successful operation.
24. Of 40 cartons of milk sitting on a grocery store shelf 4 have turned sour. If 5
cartons are chosen at random
a) Find the probability that no sour cartons are chosen.
b) How many would you expect to be sour of the 5 chosen cartons?
25. If a paratrooper has a 20% chance of injury on any one jump, what is the
probability of only one injury in 5 jumps?
26. The odds against the Maple Leafs winning a hockey game are 7 to 1. Find
your net expected winnings if you win $10 if they win and must pay $1 if they lose
or tie.
27. A package contains 7 brown, 3 green, 2 yellow, 6 red and 9 orange candy
coated peanuts. Two peanuts are chosen at random without replacement. Find
the probability of choosing:
a)
i) two red
ii) two of the same colour
iii) a red and a green
b) Construct a probability distribution table and a relative frequency graph
illustrating the probability distribution of the random variable, R, the number of
red candy coated peanuts.
Unit 5: Statistics of One Variable
28. The following data are given:
Shoe
sizes:
9
9
6
9
9
8
7
7
8
7
7
12
6
6
10
8
7
10
9
6
11
9
7
9
11
8
8
7
9
7
9
Calculate:
a) the mean, median, mode
b) the range, Q1, Q2, Q3 and IQR
c) standard deviation, variance
d) In which percentile is a person with a shoe size of 9?
e) Determine if there are any outliers
f) Illustrate this data using a modified Box and Whisker Plot.
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MDM4U – Final Exam Review
29. Describe the type of sample used in each of the following scenarios.
a) A proportionate number of boys and girls are randomly selected from a class.
b) A software company randomly chooses a group of schools in a particular
school district to test a new timetable program.
c) A newspaper prints a questionnaire and invites its readers to mail in their
responses.
d) A telephone survey company uses a random number generator to select which
household to call.
e) An interviewer polls people passing by on the street.
30. What are the three measures of central tendency? Describe what each
calculates.
Unit 6: Statistics of Two Variable Data
31. In Mr. Strong’s fitness course, several fitness scores were taken. The following
sample is the number of push-ups and sit-ups done by 10 random selected
students.
Student
Push-ups, x
Sit-ups, y
1
2
3
4
5
6
7
8
9
10
27
22
15
35
30
52
35
55
40
40
30
26
25
42
38
40
32
54
50
43
a) Create a scatter plot correctly labeled.
b) Use the calculator to determine the line of best fit.
c) Determine and interpret the correlation coefficient. What type of correlation
have you found?
d) If a student performs 100 push-ups, predict how many sit-ups they should be
able to do. What term is used when we describe what you just did in predicting
this value in this range?
e) What terms is used when we predict a value outside of the data range?
32. An engineer is testing a transmitter at a new radio station. He has measured
the radiated power at various distances from the transmitter. The engineer’s
readings are in microwatts per square meter.
Distance (km)
2.1
4.5
8.1
10.0
11
12
13
15
20.5
Power Level ( W / m 2 )
505
70
30
19
15
14
12
9
4
a) Create a scatter plot of this data.
b) Determine the equation for the best curve of best fit for this data and place the
curve that you found on your graph. What type of equation is this? Why is it
superior to the other types of curves that you could have used?
c) What is the coefficient of determination for your curve of best fit? What is the
meaning of this value?
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MDM4U – Final Exam Review
Unit 7: Normal Distribution
33. It is found that a certain brand of light bulb has a lifetime mean of 215 hours
with a standard deviation of 12 hours. The lifetime hours are normally distributed.
What percentage of the bulbs would you expect for each of the following? Include
a rough sketch of the normal distribution curve (with z values) that applies to
each of the questions.
a) to fail in less than 200 hours
b) to have a lifetime of more than 250 hours
c) to have a lifetime of exactly 228 hours
34. The probability that Julia tips her canoe on any trip is 35%.
a) Using binomial distribution, find the probability that Julia falls out exactly 15
times in 45 trips.
b) Using normal approximation to the binomial distribution, find the probability
that Julia falls out 15 times in 45 trips. Compare the results from the normal
approximation with the results from the calculations using a binomial
distribution in part a.
c) State why the normal approximation is appropriate to use in this situation.
d) Find the probability that Julia falls out less than 15 times in 45 trips.
Answers:
1. a) 11990 b) 715
2. a) 17,576,000 b) 6,955,200
3. a) 10 b) 5 c) 31
4. a) 1440
5. a) 720 b) 6,720 c) 415,800
6. a) b) 44% c) 24.5%
7. a) 9 ways b) 34 routes
8. a) b) 32
9. a) b) 185 c) 71
10. a) 1 b) 4 c) 6 d) 5 e) 15
11. 65,780
12. 1200
13.
14. 19,008,000
15. a) 1287 b) 420 c) 1266 d) 531 e) 27
16. (n  1)( n  2)
2
17. a) dependent b) mutually exclusive c) independent d) non-mutually exclusive
18. 1/90
19. a)
b) i).0.25 ii) 0.5 iii) 0.125 iv) 0.5 v) 0.875 vi) 0.5
20. a) 1/6 b) 1/6
21.a) 0.000742 b) same colour more likely (see solutions)
22. a) b) c) $7.45 d) not fair game
23. 42%
24. a) 57.3% b) 0.5
25. 41%
26. 38 cents
27. a) i) 4.3% ii) 21.7% iii) 5.13% b) see solutions
28. a) mean=8.21 median=8 mode=9 b) Q1=7, Q2=8, Q3=9, IQR=2 c) s.dev=1.57, var=2.47
d) 70th percentile e) 12 is an outlier f) see solutions
29. a) stratified b) cluster c) voluntary d) simple random e) convenience
30.
31. a) b) y=0.66x+14.91 c) r=0.84 strong positive correlation d) 81 sit-ups
e) extrapolation
32.
33. a) 10.56% b) 0.18% c) 2%
34. a) 12.19% b) 11.98% c) np>5 & nq>5 d) 34.83%
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