Math 12/ Advanced Math 12 Probability Unit
Time frame: 10%-15% of total course time is suggested by the Department of Education.
1
We recommend 2 - 2 weeks at the beginning of the semester. Then include this topic
2
in the cumulative testing.
Suggestions:
Topic
Definition of probability
Fundamental Counting
Principle
Suggestions
Focus A (pp.300-302)
Emphasize definition of
probability, complement ( A is the
complement of A ), experimental
vs. theoretical probability
*Some teachers do the Venn
Diagrams before this principle.
Inv.3 (pp.307-309)
Explain tree diagrams.
Emphasize: “and” indicates
multiplication
Venn Diagrams
Inv.5 (not mutually exclusive)
Inv.6 (mutually exclusive)
Emphasize:
“and” indicates intersection
”or” indicates union
Recommended
questions
#’s 3,4(omit 4e),
5,6,10,11
p.318, #43
See supplementary
Questions.
Page 308 #’s 3-7
Page 321 #49,50
See supplementary
Questions.
Pages 312-315 #’s
21-36
Omit #35 at this time.
See supplementary
Questions.
Page 301 # 4(e)
P(AorB)=P(A)+P(B)-P(A and B)
P(A and B)= P(A  B)
For mutually exclusive events,
P(A and B)=0 (no intersection)
P(AorB)=P(A)+P(B)
Arrays
An array (table) is a useful way to
explore the possible sums when
tossing two dice.
Factorial Notation
Focus F (pp.329-331)
Combinations/Permutations Inv.10 (pp.327-328) to emphasize
the distinction between
combinations and permutations.
Page 315 # 35
#’s 8-14, omit #12
p.328,#’s1,2
See supplementary
Questions.
Permutations
Special Permutations
Combinations
Emphasize methods of calculating
permutations:
1) list all possible outcomes
2) fundamental counting
principle
3) formula
How do we know when to use
permutations?
-indication of order or position
Ex:
-1st, 2nd, 3rd,
- President., Vice Pres., Treasurer
- arrangements
1) circular (n-1)!
Ex.: How many ways can eight
people be arranged around a
circular table?
2) Repetitions within
permutations
Ex.: How many permutations are
there using the letters in the word
bubbles?
When do we use combinations?
-indication that order, position
not important
Ex. Anonymous groups,
committees
Applying
Section 5.4, Focus H page 336
Combinations/Permutations Emphasize nature of the question:
to Probability
“How many ways…?” (not a
probability question; requires a
whole number answer) vs.
“What is the probability that…?”
(requires a rational number answer
between 0 and 1).
Conditional Probability
Focus E, Investigation 8
(Advanced only)
Emphasize , as in Venn Diagrams,
“and” in Conditional Probability is
intersection
Recall A is the complement of A.
Page 332
#16,17,18a,19
See supplementary
Questions.
Page 332 #18b,20
See supplementary
Questions.
Page 333 #21-34
See supplementary
Questions.
Page337 #2-16
Page 310 #10,11,12
Page 324 # 58,59,60
See supplementary
Questions.
Page 320 #47,48
Page 322 Procedure
A,B,C

Omit: Investigation 1 page 303, Investigation 2page 304, Focus D,
Investigation 7, Sections 5.5,5.6

Caution:
Many students have difficulty in probability problems involving the words
“or” and “and”. It is important to make clear the use of “or” in English language in
comparison to its use in Mathematics. “Or” in English refers to either/or.
“Or” in Mathematics includes the possibility of both.
In Venn Diagrams and Conditional Probability, “and” implies intersection.
In other situations, “and” implies multiplication (according to the Fundamental
Counting Principle). An extension of this principle is P(A and B)= P(A) x P(B).

Students should be using the Study Guide from the Department of Education

Websites
Nova Scotia Department of Education http://itembank.ednet.ns.ca
Newfoundland Dept. of Education
http://www.ed.gov.nl.ca/edu/k12/pub/sample.htm

Yearly timeline 2005-2006 used by Elaine MacEachern, GBHS
First semester
Sept.9-Sept.26: Probability Chapter5
Sept.27-Oct.27: Circles Chapter 4
Oct.31-Dec.6: Quadratics Chapter 1
Dec.7-Dec.16; Jan.3-Jan.24: Exponential Growth
Second semester
Feb.8-Feb.23: Probability
Feb.24-March28: Circles
March29-May4: Quadratics
May5-June12: Exponential Growth
Questions from Nova Scotia Provincial Mathematics Examinations
1) Two coins and one die are simultaneously tossed on a table. What is the probability of
obtaining two heads and a six?
a) 5 / 12 (b) 1/ 24 (c) 1/ 5 (d) 1/ 8
2) If we shuffle a standard deck of 52 cards and randomly select one card, what is the
probability of selecting a king or a queen?
a) 1/ 52 (b) 1/ 26 (c) 1/ 13 (d) 2/ 13
3) Two boxes contain marbles. The first contains five red marbles and three white
marbles. The second box contains four black marbles and seven green marbles. One
marble is chosen at random from each box. What is the probability that a white and a
black marble will be chosen?
a) 3/ 22 (b) 33/ 32 (c) 1/ 12 (d) 12/ 35
4) Tim and Rebecca are the first and second students in a line of 7 students waiting to buy
tickets for a concert. The number of different orders in which the remainder of the
students can line up behind them is
7!
a) 5! (b) 7! (c) (5!)(2!) (d)
2!
5) Use a real-life example to explain why 4 C4  1 .
6) A fair coin is tossed three times. Draw a tree diagram and determine the probability of
obtaining
a) no heads
b) exactly 2 heads
c) at least one head
7) A class is made up of 13 girls and 9 boys. If 5 students are chosen at random, what is
the probability that 5 girls will be chosen?
8) A bag contains x red beads, y yellow beads and z blue beads. Two beads are drawn at
random, with the first being replaced before the second is drawn. What is the probability
(in terms of x, y and z) that both beads are red?
2x
x2
x
2x
a)
(b)
(c)
(d)
2
( x  y  z) 2
x yz
x yz
( x  y  z)
9) A class is made up of 13 girls and 9 boys. What is the probability that, of 5 students
chosen at random, 3 are girls and 2 are boys?
10) While working on a problem, Christine observed that 5 P1 and 5 C1 gives the same
value, but that the value for 5 P2 is larger than the value for 5 C 2 . Why is this so?
11) Mr. Smith has 3 pairs of black pants and 2 pairs of grey pants in the first drawer of
his dresser. In the second drawer, he has 1 white shirt and 4 multi-colored shirts. He gets
up late one morning and without looking quickly grabs a pair of pants from the first
drawer and a shirt from the second drawer. What is the probability that he grabs a pair of
black pants and a white shirt?
a)
3
25
(b)
4
25
(c)
4
5
(d)
2
5
12) Based on the information in the table below, the probability that a person did not
take vitamins regularly given that this person developed a cold is
Developed a cold
Took vitamins
regularly
Did not take
vitamins
regularly
Total
a)
115
352
(b)
115
167
Did not develop
a cold
Total
46
139
185
115
52
167
161
191
352
(c)
46
352
(d)
115
161
13) Empire Theatres observed 25 movie goers. Of them, 15 ordered a drink, 8
ordered popcorn and 5 ordered both.
a) Display the results in a Venn Diagram.
b) Based on your diagram, what is the probability that a movie-goer will buy
popcorn, given that the person buys a drink?
14) Use the following chart to calculate the probability of being a blonde, given you are
male,
Blonde
Not Blonde
Male
15
18
Female
8
12
15) Peter, Mary and Susan are part of a group of 10 people. An executive consisting
of Peter as the President, Mary as the Treasurer, and Susan as the Secretary could
be formed from this group. What is the probability this executive will be formed?
1
1
3!
3
a)
(b)
(c)
(d)
P
10 P3
10 C 3
10 C 3
10 3
16)
Number of students
Applied for a summer job
120
Number of students
got an interview
100
Number of students
got a summer job
75
Number of students
did not get an interview
20
Number of students
did not get a summer job
25
What is P( student got a summer job / student got an interview) ?
1
a) 5
(b) 3
(c) 5
(d)
8
4
6
3
17) You want to put 8 different books on a shelf, side by side. In how many ways can
these books be arranged?
8!
a) 8! b)
c) 8 P1
d) 8 C 8
2!
1000!
is:
999!
(b) 1.001
(c) 1000
18) The value of
a) 1
(d) undefined
19) A Prom Committee of 7 will be chosen from 10 boys and 12 girls. Calculate the
probability that exactly 5 girls will be chosen to be on the committee.
20) Use a real-life example to explain why 5 C 2  5 C3 .
21) At a high school, the following data was obtained indicating students enrolled in
grade 12 math.
Sex
Advanced Math 12
Math 12
Total
Male
63
85
148
Female
72
112
184
Total
135
197
332
Assume that event A is “enrolled in Advanced Math 12” and event B is “male”.
(a) P(A)
(b) P(B/A)
22) If an event can succeed in “s” ways and fail in “f” ways, then the probability of
success is:
s
s
a)
(b)
(c) ( s)( f )
(d) 1  f
f
f s
23) Two dice are thrown. Given that the sum of the two numbers on the pair of dice is
greater than 7, what is the probability that these two numbers are the same?
1
5
1
1
a)
(b)
(c)
(d)
5
12
12
2
24) In a school of 200 students, 80 have blood type O. If 5 students are chosen at
random, what is the probability of selecting five students with type O blood?
1 1 1 1 1
x x x x
a)
80 79 78 77 76
5
b)
80 C 5
5
c)
80 P5
C
d) 80 5
200 C 5
25) Refer to the following chart
Event A
10
15
Event B
Event B
Event A
20
25
Calculate
a) P(A or B)
b) P(A/B)
26) From a group of 5 men and 6 women, what is the probability that a committee
formed at random will consist of 3 men and 3 women?
27) Joe, Mary and George are among the seven finalists for a random draw to win
three different prizes. What is the probability that Joe will win 1st prize, Mary will
win 2nd prize, and George will win 3rd prize? Express your answer in fraction
form.
28) John, Amy and Fred tried to solve the following problem:
In a certain city, during a person’s lifetime the probability of having diabetes is 0.10
and the probability of having cancer is 0.05. What is the probability of a person’s
having either diabetes or cancer in his/her lifetime?
Suppose that event C is “person having cancer” and event D is “person having
diabetes”.
Their proposed solutions are as follows:
John’s solution: P(C and D)= (0.10)(0.05)=0.005
Amy’s solution: P(C or D) = 0.10+0.05=0.15
Fred’s solution: P(C or D) = 0.10+0.05-0.005 + 0.145
a) Which student has the correct answer?
b) Explain why the other two solutions are NOT correct.
500!
is
499!
(b) 1.002
(c) 500
29) The value of
a) 1
(d) undefined
30) Consider the following Venn diagram
Event K
Event L
Which of the following is correct?
a) P(K or L) = P(K) + P(L)
b) P(K and L) = P(K) + P(L)
c) P(K or L) = P(K) x P(L)
d) P(K and L) = P(K) x P(L)
30) Your math teacher gives your class a list of eight questions to study. Five of the
eight questions will be randomly selected for the next test. If you study only the
first five questions from the list, the probability that all of those five questions
will be on the test is:
1
1
5
1
a)
(b)
(c)
(d)
8
8
8 C5
8 P5
31) There are 5 red marbles and 7 blue marbles in a bag. Two marbles are chosen
randomly without replacement. The probability that a red and then a blue marble
will be chosen is:
1
1
5 7
5 7

a)
x
(b)
(c)
(d)
12 11
12 11
5 C1 x 7 C1
12 C 2
32) There are 15 jellybeans distributed in a jar: 5 are yellow and 10 are orange. You
reach into the jar and, without looking, remove 2 jellybeans. What is the
probability that you will remove 2 yellow jellybeans?
33) In a group of 15 people, 4 are left-handed and 11 are right-handed. Seven people
are selected at random from this group.
(a) What is the probability that all 4 left-handed people will be selected?
(b) If Sarah and Mike, two of the left-handers, have already been chosen, what is the
probability that all the other members selected will be right handed?
34) Create a real-life problem that demonstrates
P(A or B) = P(A) + P(B) – P(A and B) when the events A and B are NOT mutually
exclusive. (You do not have to solve the problem)
Teacher Resource
Probability Unit
Mathematics 12/ Advanced Mathematics 12
Contributors: Arlene Andrecyk
Elaine MacEachern
Sandy Urquhart
Cape Breton-Victoria Regional School Board
January 2007.