Christ the Redeemer Catholic Schools
Mathematics 30-2
Course Review Package
Mrs. Sample
2014
Fun Facts about the Diploma Exam
How long is the exam?
The exam is designed to be completed in 2.5 hours; you are allowed an addition half hour if you need it.
How many questions are there on the diploma exam?
There are 28 multiple choice questions and 12 numerical response questions.
What is the breakdown of the exam in terms of topics learned in class?
50% of the exam (approximately 20 questions) will cover topics such as rational expressions, rational
equations, polynomials, exponential functions, logarithmic functions, and sinusoidal functions.
33% of the exam (approximately 13 questions) will cover topics such as the fundamental counting principle,
permutations, combinations, and probability.
17% of the exam (approximately 7 questions) will cover topic such as logical reasoning, problem solving
strategies, and set theory.
Numerical Response Questions
You should be prepared to see three different styles of numerical response questions on the diploma. It is important to
note that for all types, responses should be aligned to the left and numerical bubbles must be filled in.
Question and Solution
When the answer to be recorded
cannot be a decimal, students are
asked to determine a whole number
value. If the answer can be a
decimal value, then students are
asked to record their answer to the
nearest tenth or nearest hundredth,
as specified in the question.
Students should retain all decimals
throughout the question and
rounding should only occur in the
final answer.
Correct-Order Question and
Solution
Any-Order Question and Solution
Rational Expressions
A rational expression is the ratio
(think fraction) of two
polynomials. They are written in
the form
1. An expression that is equivalent to
A. 𝑥 + 1, 𝑥 ≠ 0
C. 2𝑥, 𝑥 ≠ 0
𝑝(𝑥)
𝑞(𝑥)
𝑥 2 +𝑥
,𝑥
𝑥
≠ 0, is
B. 𝑥 2 + 1, 𝑥 ≠ 0
D. 𝑥 2 , 𝑥 ≠ 0
2. Numerical Response The non-permissible value of 𝑥 in the expression
2𝑥+1
is _______.
3𝑥−9
Non-Permissible Values:
2𝑥+4
Since division by zero is
impossible within the laws of
mathematics, the denominator of a
rational expression may never be
equal to zero. Therefore, any
values of the variable that make the
denominator equal to zero are nonpermissible.
*When working with division, you
must also look in the numerator
of the second expression!
3. When the rational expression 𝑥 2 −4 is simplified, the equivalent expression is
A.
C.
2
,𝑥
𝑥−2
2
,𝑥 ≠
𝑥
≠ −2, 2
B.
−2, 0, 2
D.
2
,𝑥 ≠
𝑥+2
2
,𝑥 ≠ 0
𝑥
−2
To simplify rational expressions:
Fully factor both the numerator
and denominator, and then cancel
out any common factors.
To multiply rational expressions:
Simplify the expressions as much
as possible. Then, multiply
together the numerators and
denominators of both expressions.
Simplify further, if possible.
4. Numerical Response One possible selection to form a correct
simplification is (𝑥 − 3), (𝑥 + 4), and (𝑥 2 − 9), so Henry records the code
159. To form another correct simplification, a code for another possibility
for
a is _______________ (Record in the first column)
b is _______________ (Record in the second column)
c is _______________ (Record in the third column)
To divide rational expressions:
Simplify the expressions as much
as possible. Then, multiply the first
expression by the reciprocal of the
second.
5. The simplified form of
A.
C.
To add rational expressions:
Simplify both expressions as much
as possible. Then, multiply the top
and bottom of the expressions to
create a common denominator, and
add the expressions together.
6. The simplified form of
A.
C.
To subtract rational expressions:
The same process as for adding,
but don’t forget to distribute the
negative!
1. A
2. 3
𝑥
𝑥−2
𝑥
𝑥+3
D.
𝑥 2 +3𝑥
𝑥 2 −4
A.
3. A
≠ −2, 0, 2
𝑥+3
D.
5𝑥+3
𝑥+2
𝑥+2
𝑥+3
𝑥+3
𝑥+2
is
≠ −2
B.
≠ −2
D.
4. 258 or 148 or 269 or 357
2(2𝑥+3)
(𝑥+2)(𝑥−2)
2(2𝑥−3)
,𝑥
(𝑥+2)(𝑥−2)
÷ 𝑥+2 , 𝑥 ≠ −3, −2, 2 is
B.
4𝑥
−𝑥−3
,𝑥
2𝑥+4
−𝑥−3
,𝑥
𝑥+2
is
B.
7. The simplified form of 𝑥+2 −
C.
Answers:
𝑥+3
(𝑥+2)(𝑥−2)
4𝑥
(𝑥+2)(𝑥−2)
𝑥
3𝑥
+ 𝑥 2 +2𝑥
𝑥 2 −4
5. D
−𝑥+3
,𝑥
2𝑥+4
−𝑥+3
,𝑥
𝑥+2
6. A
≠ −2
≠ −2
7. C
Rational Equations
1. The solution to
A. 𝑥 =
3
2
5𝑥−1
4𝑥+11
3
4
= is
B. 𝑥 =
3
37
8
C. 𝑥 =
37
32
D. 𝑥 =
29
8
C. 𝑥 =
9
35
D. 𝑥 =
9
25
5
2. The solution to x + 3 = 10 is
1
A. 𝑥 = 5
B. 𝑥 =
4
3
5
6x
3. The solution to +
= 6 is
x
𝑥+1
1
A. 𝑥 = 2
B. 𝑥 =
3
C. 𝑥 = −1
2
D. 𝑥 = 3
You can solve a rational equation
algebraically by multiplying each
term in the equation by the lowest
common denominator (LCD) to
eliminate fractions from the
equation. Then solve the resulting
linear or quadratic equation.
If a root of a rational equation is a
non-permissible value of the
rational expressions in the original
equation, it is an extraneous root
and must be dismissed as a valid
solution.
Each solution to a rational
equation can be verified by
substituting it into the original
equation. If the left side equals the
right side, then it is a valid
solution to a rational equation in a
non-contextual problem.
4. In which step did the student make their first error?
A. Step 1
B. Step 2
C. Step 3
2𝑥
𝑥
D. Step 4
18
5. The solution to
+
= 2 is
𝑥+3
𝑥−3
𝑥 −9
A. 𝑥 = 3
B. 𝑥 = −2, 3
C. 𝑥 = −3, 2
D. 𝑥 = −2
6. What is the correct equation to model this situation?
160
160
160
160
A. 𝑥−40 + 𝑥 = 1.6
B. 𝑥−40 − 𝑥 = 1.6
C.
Answers:
160
𝑥−40
= 1.6
1. B
D.
2. D
3. A
160
𝑥
4. A
= 1.6
5. D
6. B
When solving a contextual
problem, it is important to check
for solutions that are valid
solutions to the equation but are
inadmissible in the context of the
problem (for example, a rectangle
cannot have a negative side
length).
Exponential Functions and Equations
From your formula sheet:
𝑦 = 𝑎 𝑥 ↔ 𝑥 = log 𝑎 𝑦
1. Which of the following exponential functions could be used to model the
value of the car, 𝑣(𝑡), after 𝑡 years?
A. 𝑣(𝑡) = 5274(1.18)𝑡
B. 𝑣(𝑡) = 29300(1.18)𝑡
C. 𝑣(𝑡) = 5274(0.82)𝑡
D. 𝑣(𝑡) = 29300(0.82)𝑡
Exponential Functions:
𝑦 = 𝑎 ∙ 𝑏𝑥
Properties of Exponential
Functions:
 no x-intercepts
 y-intercept of a
 curve extends from
quadrant II to quadrant I
 domain {𝑥|𝑥 ∈ 𝑅}
 range {𝑦|𝑦 > 0, 𝑦 ∈ 𝑅}
2. How many times as intense is an earthquake of magnitude 8.5 than an
earthquake with a magnitude of 6.9?
A. 10
B. 20
C. 30
D. 40
3. A painting was purchased in 2012 for $10 000. If the painting appreciates in
value at 5%/a, then which of the following graphs best models the value of
this painting for the next 40 years?
A.
B.
To estimate the solution to an
exponential equation, you can enter
the exponential equation as a
system of equations on a graphing
calculator and graph the system.
To determine an exact solution to
an exponential equation
algebraically, write both sides of
the equation as powers with the
same base (if possible). Then set
the exponents equal to each other,
and solve the resulting equation.
C.
Financial Applications
The exponential function that
models compound interest is
𝐴(𝑛) = 𝑃(1 + 𝑖)𝑛 . Remember,
𝑖=
D.
4. Solve 2𝑥−1 = 3𝑥−2 .
A. 𝑥 = 1.0
B. 𝑥 = 1.3
𝑎𝑛𝑛𝑢𝑎𝑙 𝑟𝑎𝑡𝑒
# 𝑡𝑖𝑚𝑒𝑠 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑖𝑠 𝑝𝑎𝑖𝑑 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟
C. 𝑥 = 3.7
D. 𝑥 = 5.4
5. Determine how long it will take for the investment to be worth at least $150
at 2.4%/a, compounded monthly.
A. 202 months
B. 203 months
C. 204 months
D. 205 months
Answers:
1. D
2. D
3. A
4. C
5. B
Logarithmic Functions and Equations
Change of Base Formula (to
evaluate logs of bases other than
10 in your calculator!)
log 𝑎 𝑐
log 𝑏 𝑐 =
log 𝑎 𝑏
Laws of Logarithms
log 𝑏 (𝑀𝑁) = log 𝑏 𝑀 + log 𝑏 𝑁
𝑀
log 𝑏 ( ) = log 𝑏 𝑀 − log 𝑏 𝑁
𝑁
log 𝑏 (𝑀𝑛 ) = 𝑛 log 𝑏 𝑀
Logarithmic Regression Functions
𝑦 = 𝑎 + 𝑏 ln 𝑥
The common logarithm has base
10, and is written as
𝑦 = log 𝑥
1. Use logarithmic regression to determine the Minimum Perceivable
Difference for a 2100 g object, to the nearest whole gram.
A. 17 g
B. 22 g
C. 27 g
D. 32 g
2. The expression log 6 3 + log 6 12 in simplified form is
A. log 6 15
B. log 6 36
C. log 6 (−9)
The natural logarithm has base e,
and is written as
𝑦 = ln 𝑥
1
D. log 6 (4)
3. Numerical Response If the concentration of hydrogen ions in the solution
is doubled, the new pH of the solution, to the nearest tenth is _______.
Sometimes, to solve an
exponential equation, it is useful to
apply the common logarithm to
both sides of the equation.
4. Which pair of equations are not equivalent?
A. log 100 = 2 and 102 = 100
B. log 2 8 = 3 and 23 = 8
C. ln 𝑥 = 2 and 102 = 𝑥
D. log 𝑎 5 = 2 and 𝑎2 = 5
5. What is 2 ln 𝑥 − ln 𝑦 expressed as a single logarithm?
𝑥2
B. ln(2𝑥 − 𝑦)
C. ln(𝑥 2 − 𝑦)
A. ln
𝑦
𝑥 2
𝑦
D. ln ( )
6. What is log 𝑃 − 4 log 𝑄 − log 𝑅 expressed as a single logarithm?
𝑃
𝑃𝑅
𝑃
𝑃𝑅
A. log (𝑄4 𝑅)
B. log (4𝑄)
C. log (4𝑄𝑅)
D. log ( 𝑄4 )
Answers:
1. C
2. B
3. 6.3
4. C
Characteristics of Logarithmic
Functions:
 x-intercept of 1
 no y-intercept
 curve extends from
quadrant IV to I or from
quadrant I to IV
 domain {𝑥|𝑥 > 0, 𝑥 ∈ 𝑅}
 range {𝑦|𝑦 ∈ 𝑅}
5. A
6. D
Polynomial Functions
Characteristics of Degree 0
(constant) Functions
No x-intercept
Domain {𝑥|𝑥 ∈ 𝑅}
Range {𝑦|𝑦 = constant, 𝑦 ∈ 𝑅}
No turning points
1. Consider the graph of 𝑓(𝑥) = −(𝑥 + 1)(𝑥 − 2)2 . Which of the following
statements are false?
A. 𝑓(𝑥) is a cubic function
B. 𝑓(𝑥) has two x-intercepts
C. 𝑓(𝑥) extends from quadrant II to quadrant IV
D. 𝑓(𝑥) has a y-intercept of 4
Characteristics of Degree 1
(linear) Functions
1 x-intercept
Domain {𝑥|𝑥 ∈ 𝑅}
Range {𝑦|𝑦 ∈ 𝑅}
No turning points
2. The rate at which snow fell on a driveway on a particular day can be
modelled by 𝑦 = −3𝑥 2 + 6𝑥, where y represents the rate of snowfall in
ft 2 /hr, and x represents the time in hours. To estimate the length of time
that snow fell on this particular day, a student should determine the
A. y-intercept
B. x-coordinate of the vertex
C. y-coordinate of the vertex D. difference between the x-intercepts
3. Which of the following graphs would most likely represent the graph of a
cubic function?
A.
B.
C.
D.
Characteristics of Degree 2
(quadratic) Functions
Number of x-intercepts
Domain {𝑥|𝑥 ∈ 𝑅}
Range 𝑦|𝑦 ≥ 𝑚𝑖𝑛𝑖𝑚𝑢𝑚, 𝑦 ∈ 𝑅} or
{𝑦|𝑦 ≤ 𝑚𝑎𝑥𝑖𝑚𝑢𝑚, 𝑦 ∈ 𝑅}
1 turning point
Characteristics of Degree 3
(cubic) Functions
May be 1, 2 or 3 x-intercepts
Domain {𝑥|𝑥 ∈ 𝑅}
Range 𝑦|𝑦 ∈ 𝑅}
May have 0 or 2 turning points
Inflection points: a curve in a
cubic function that is not a turning
point
4. The new box must hold 1000 mL of juice. Determine the largest amount by
which each dimension of the juice box can be increased, to the nearest tenth
of a centimeter.
A. 2.7 cm
B. 3.5 cm
C. 4.2 cm
D. 5.3 cm
Answers:
1. D
2. D
3. B
4. B
Sinusoidal Functions
From your formula sheet:
𝑦 = 𝑎 ∙ sin(𝑏𝑥 + 𝑐) + 𝑑
Period:
1. Numerical Response The height of the pendulum at the moment of release,
to the nearest tenth of an inch, is _________ in.
2𝜋
𝑏
Angles can be measured in both
degrees and radians.
𝜋 radians = 180°
Periodic function: a function
whose graph repeats in regular
intervals or cycles
2. According to the sinusoidal regression function, the maximum height of the
rider above the ground is
A. 2 m
B. 6 m
C. 8 m
D. 14 m
3. Numerical Response When the rider is at least 11.5 m above the ground,
she can see the rodeo grounds. During each rotation of the Ferris wheel, the
length of time that the rider can see the rodeo grounds, to the nearest tenth
of a minute, is _________ min.
The height of the Ferris wheel can be represented by a sinusoidal function of the
form ℎ = 𝑎 sin(𝑏𝑡 − 1.57) + 𝑑.
4. What is the value of a in this function?
A. 0.13
B. 8
C. 10
D. 48
5. What is the value of b in this function?
A. 0.13
B. 8
C. 10
D. 48
6. What is the value of d in this function?
A. 0.13
B. 8
C. 10
D. 48
Answers:
1. 3.3
2. D
3. 1.8
4. B
5. A
6. C
Midline: the horizontal line
halfway between the maximum
and minimum values of a periodic
function
𝑚𝑎𝑥 + 𝑚𝑖𝑛
𝑚𝑖𝑑𝑙𝑖𝑛𝑒 =
2
Amplitude: the distance from the
midline to either the maximum or
minimum value of a periodic
function; the amplitude is always
expressed as a positive number.
𝑚𝑎𝑥 − 𝑚𝑖𝑛
𝑎𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒 =
2
Period: the length of the interval
of the domain to complete one
cycle.
2𝜋
𝑝𝑒𝑟𝑖𝑜𝑑 =
𝑏
Logical Reasoning
Good problem solving strategies
include:
 Guess and check
 Look for a pattern
 Make a systematic list
 Draw or model
 Eliminate possibilities
 Simplify the original
problem
 Work backward
 Develop alternative
approaches
Examples
1. The fifth row of the pattern will be
A. 1 234 × 8 + 4 = 9 876
B. 1 234 × 8 + 5 = 9 876
C. 12 345 × 8 + 4 = 9 876
D. 12 345 × 8 + 5 = 98 765
2. NUMERICAL RESPONSE It is Margaret’s turn, and she determines that
she can guarantee a win by placing the letter M in
Row __________ Column __________
3. Which of the following pictures is next in the pattern?
A.
B.
C.
Answers:
1. D
2. 42
D.
3. A
Set Theory
Some important terms,
symbols and definitions:
1. The number of students in the universal set is
A. 61
B. 64
C. 78
D. 98
𝑈
Universal set: a
set of all the
elements under
consideration for
a particular
context (also
called sample
space).
𝐴′
Complement: all
the elements of a
universal set that
do not belong to a
subset of it.
∅
Empty set: a set
with no elements;
for example, the
set of odd
numbers divisible
by 2 is the empty
set.
∩
Intersection: the
set of elements
that are common
to two or more
sets.
⊂
Subset: a set
whose elements
all belong to
another set.
∪
Union: the set of
all the elements in
two or more sets.
2. Numerical Response The number of students taking Art is _______.
3. Numerical Response The number of students not taking Math is _______.
4. The number of students taking Math or Art is
A. 17
B. 61
C. 78
D. 98
5. The number of people who saw “The Princely Groom” is
A. 17
B. 61
C. 78
D. 98
6. Numerical Response The number of people who saw “Metal Man” and “The
Princely Groom” but not “Quick and Angry” is _______.
7. The number of people who saw “Metal Man” only is
A. 17
B. 61
C. 78
D. 98
8. The number of people who saw “Metal Man” or “Quick and Angry” is
A. 17
B. 61
C. 78
D. 98
9. Which of the following statements is true for sets 𝑅, 𝑆 and 𝑇?
A. 𝑅 ⊂ 𝑆
B. 𝑅 ⊂ 𝑇
C. 𝑆 ⊂ 𝑅
Disjoint sets: two
or more sets
having no
elements in
common.
n(A) Notation,
describing the
number of
elements in set A.
D. 𝑇 ⊂ 𝑅
10. Which of the following statements is not true for sets 𝑅, 𝑆 and 𝑇?
A. 𝑇 ⊂ (𝑅 ∩ 𝑆)
B. T⊂ (𝑅 ∩ 𝑇)
C. (𝑅 ∩ 𝑆) ⊂ 𝑇
D. (𝑅 ∩ 𝑇) ⊂ 𝑇
Answers:
1. D
2. 50
3. 53
4. C
5. C
6. 10
7. B
8. C
9. D
10. C
Fundamental Counting Principle
1. Determine the number of six-digit odd numbers that can be created using
the digits 0 to 9 without repetition.
A. 60480
B. 67200
C. 75600
D. 151200
𝑛! = 𝑛(𝑛 − 1)(𝑛 − 2) … 3 ∙ 2 ∙ 1
The Fundamental Counting
Principle: if there are a ways to
perform one task and b ways to
perform another, then there are 𝑎 ∙
𝑏 ways of performing both.
2. Determine the number of distinct arrangements of all the letters in the word
TATTOO.
A. 30
B. 60
C. 120
D. 720
The Fundamental Counting
Principle can be extended to more
than two tasks.
The Fundamental Counting
Principle does not apply when
tasks are related by the word OR.
3. Determine the number of 3-letter arrangements of the letters of the word
DIPLOMA.
6. 7𝐶3
7. 7𝑃3
8. 7! 3!
9. 7!
4. Only six people have tickets for 2 prizes in a school draw, and each person
has only one ticket. Once a ticket is drawn for a prize, it is not reentered in
the draw. What is the probability that Bill wins the first prize and Mary
wins the second prize?
1
A.
B.
C.
D.
15
1
18
1
30
1
36
5. Numerical Response How many different flight options can Julian select
from if he flies with this airline? ________
Answers:
1. B
2. B
3. B
4. C
5. 8
Permutations
From your formula sheet:
𝑛𝑃𝑟
1. Numerical Response For each context, use a 1 to indicate the problem
should be solved using a permutation and use a 2 to indicate the problem
should be solved using a combination.
=
𝑛!
(𝑛 − 𝑟)!
Permutation: an arrangement of
distinguishable objects in a
definite order, where each object
appears only once in each
arrangement.
In the case of repeated identical
objects:
𝑛!
𝑎! 𝑏! 𝑐!
2. Numerical Response A 7-player volleyball team must stand in a straight
line for a picture. How many different arrangements can be made for the
picture?
3. Numerical Response Determine the number of arrangements for the
volleyball team picture that can be made for the picture if the tallest player
must stand in the middle.
Answers: 1. 1221
2. 5040
3. 720
Combinations
From your formula sheet:
𝑛!
𝑛𝐶𝑟 =
(𝑛 − 𝑟)! 𝑟!
𝑛𝐶𝑟
1. In a group of 9 students, there are 4 females and 5 males. How many
different 4-member committees have 2 females and 2 males?
A. 24
B. 60
C. 126
D. 3024
𝑛
=( )
𝑟
When order does not matter in a
counting problem, you are
determining combinations.
2. In a group of 9 students there are 4 females and 5 males. How many
different committees consist of 2 or 3 students?
A. 120
B. 156
C. 540
D. 576
From a set of n different objects,
there are always fewer
combinations than permutations
when selecting r of these objects.
When all n objects are being used
in each combination, there is only
one possible combination.
3. Ralph knows that there are 15 distinguishable possibilities when 2 people
are chosen to form a committee from a particular group of n people. How
many people were in the group?
A. 𝑛 = 4
B. 𝑛 = 5
C. 𝑛 = 6
D. 𝑛 = 7
Solving Counting Problems
When solving counting problems,
you need to determine if order
plays a role in the situation. Once
you have determined whether a
problem involves permutations or
combinations you can also use
these strategies:
 Look for conditions.
Consider these first as you
develop your solution.
 If there is a repetition of r
of the n objects to be
eliminated, it is usually
done by dividing by r!
 If a problem involves
multiple tasks that are
connected by the word
AND, then the
Fundamental Counting
Principle can be applied;
multiply the number of
ways that each task can
occur
 If a problem involves
multiple tasks that are
connected by the word OR,
add the number of ways
that each task can occur.
Answers:
1. B
2. A
4. Numerical Response The values of w, x, y, and z are _______, _______,
_______, and _______, respectively.
5. Numerical Response Triangles can be formed in an octagon by
connecting any 3 of its vertices. Determine the number of different triangles
that can be formed in an octagon.
3. C
4. 9372
5. 56
Odds and Probability
Practice Problems 1, 2, 3, 4, 5
Probability: 𝑃(𝐴) =
favourable outcomes
1. The odds in favour of the Renegades winning the season final in the football
league are listed at 10:7. The odds against the Renegades winning the season total possible outcomes
final are
Odds express a level of confidence
A. 3:7
about the occurrence of an event.
B. 3:10
C. 7:10
Odds in favour of event A:
D. 10:3
𝑃(𝐴): 𝑃(𝐴′ )
2. Statistics show that 6 out of 25 car accidents are weather-related. The odds
that a car accident is weather-related can be expressed in the form a:b. The
values of a and b are, respectively, _______ and _______.
3. A class of 35 students has 17 males. One student will be selected at random
from the class. What are the odds against the randomly selected student
being a male?
A. 17:18
B. 18:17
C. 17:25
D. 18:25
4. What is the probability of winning the Flim’em game?
A. 0.25
B. 0.33
C. 0.67
D. 0.75
Answers:
1. C
2. 619
3. B
4. A
Odds against event A:
𝑃(𝐴′ ): 𝑃(𝐴)
Mutually Exclusive Events
Mutually exclusive events: two or
more events that cannot occur at
the same time. For example, rolling
a 3 and rolling a 4 on a die are
mutually exclusive events.
You can represent the favourable
outcomes of two mutually
exclusive events, A and B, as two
disjoint sets.
You can represent the probability
that either A or B will occur by the
following formula:
𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵)
You can represent the favourable
outcomes of two non-mutually
exclusive events, A and B, as two
intersecting sets.
You can represent the probability
that either A or B will occur by this
formula:
𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) −
𝑃(𝐴 ∩ 𝐵)
1. Numerical Response If one ball is randomly selected from the box, what
is the probability that the number written on it is divisible by 3 or is an even
number?
2. A particular traffic light at the outskirts of a town is red for 30 s, green for
25 s, and yellow for 5 s in every minute. When a vehicle approaches the
traffic light, the probability that the light will be red or yellow is
7
1
1
1
A. 12
B. 2
C. 12
D. 24
3. Numerical Response In a non-leap year of 365 days, the average number
of days of the year that a tourist could expect to experience weather other
than sunshine, to the nearest whole number, is _________ days.
4. Numerical Response From the list above, the two events that are mutually
exclusive are numbered _______ and _______.
5. What is the probability that a person chosen at random from the population
watches TV at least once a day or uses a computer at least once a day?
A. 0.65
B. 0.75
C. 0.85
D. 0.95
6. Numerical Response Determine the probability of Deborah getting hit in
a game.
Answers:
1. 0.65
2. A
3. 40
4. 23 or 32
5. D
6. 0.50
Conditional Probability
Dependent events: events whose
outcomes are affected by each
other. For example, if two cards
are drawn from a deck without
replacement, the outcome of the
second event depends on the
outcome of the first event.
1. In the first experiment, Event X and Event Y are ___i___, and in the second
experiment, Event X and event Y are ___ii___.
The statement above is completed by the information in row
Row
i
ii
A.
dependent
independent
B.
dependent
dependent
C.
independent
independent
D.
independent
dependent
2. Numerical Response A box contains 6 blue balls and 4 red balls. Two
balls are drawn from the box, one after the other, without replacement. The
probability, to the nearest hundredth, that the first ball drawn is blue and the
second ball drawn is red is _________.
3. Based on previous performance, the probability of a particular baseball team
4
winning any game is . The probability that this team will win their next two
5
games is
1
4
1
16
A.
B.
C.
D. 25
5
5
25
4. What is the probability that the team will win 1 game and lose 1 game
during the next 2 games?
A. 0.16
B. 0.32
C. 0.64
D. 0.80
If event B depends on event A
occurring, then the conditional
probability that event B will occur,
given that event A has occurred
can be represented as
𝑃(𝐴 ∩ 𝐵)
𝑃(𝐵|𝐴) =
𝑃(𝐴)
or
𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴)𝑃(𝐵|𝐴)
A tree diagram is often useful for
modelling problems that involve
dependent events.
Conditional probability: the
probability of an event occurring
given that another event has
already occurred.
Independent events: If the
probability of event B does not
depend on the probability of event
A occurring, then tehse events are
called independent events. For
example, tossing tails with a coin
and drawing the ace of spades
from a standard deck of 52 playing
cards are independent events.
The probability that two
independent events, A and B, will
both occur is the product of their
individual probabilities:
𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴)𝑃(𝐵)
5. Numerical Response Assuming that having a home video game system
and having a TV in their bedroom are independent, determine the
probability, to the nearest hundredth, that a student in the class has a TV in
their bedroom.
Answers:
1. D
2. 0.27
3. D
4. B
5. 0.74
A tree diagram is often useful for
modelling problems that involve
independent events.