# Slide 1 - Thames Valley District School Board

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```					4.3 Probability Using Set
Notation
TERMINOLOGY
Venn Diagram ≡ A diagram in which
sets are represented by shaded or
coloured geometrical shapes
Compound Event ≡ Consists of two or
more simple events
Subset ≡ A set whose members (values)
are all members of another set
4.3 Probability Using Set
Notation
Intersection of Sets ≡ Elements, members,
or values that are common to both (all) sets
that intersect
   A ∩ B = {elements in both A and B}
   Set of common elements to A and B
   Ex. – Some students take French and some take English while
some take both French & English.

English

French

French & English
4.3 Probability Using Set
Notation
Union of Sets ≡ Set formed by joining or
combining all elements from set A and all elements
from set B
 A U B = {elements in A or B}

 Set formed by combining elements in A or B

   Ex. – Some students take French and some take English while some
take both French & English.

region represent?
4.3 Probability Using Set
Notation
Disjoint Sets ≡ occur when sets have no
elements in common and their
intersection is the empty set (Φ )
   n(A∩B)=Φ
   Ex. Some students take English and some take French. No one
takes both A ∩ B = Φ

These sets are A.K.A.
Mutually Exclusive
events
4.3 Probability Using Set
Notation
Additive Principle for Unions of Two Sets
   To count the number of elements in two groups or
sets that have common elements, you must
subtract the common elements so that they are
not included twice
   For sets A and B, the total number of elements in
the union of A & B is the number in A plus the
number in B minus the number in both A and B
   n(A or B)=n(A)+n(B)-n(A and B)
   n(A U B)=n(A)+n(B)-n(A ∩ B)
4.3 Probability Using Set
Notation
   Why must we subtract n(A ∩ B) in the
general formula?
   Under what circumstances would
n(A U B)=n(A) + n (B)?
4.3 Probability Using Set
Notation
Using Venn Diagrams to Solve Counting Problems
Example Of the 140 grade 12 students at SFBSS, 52
have signed up for biology, 71 for chemistry, and
40 for physics. The science students include 15
taking both biology and chemistry, 8 taking
chemistry and physics, 11 taking biology and
physics, and 2 who are taking all three.
 How many are not taking any science courses?

 Illustrate the enrolments with a Venn Diagram.
4.3 Probability Using Set
Notation
   B≡Biology, C≡Chemisty, P≡Physics
n(total)=n(B)+n(C)+n(P)-n(B ∩ C)-n(C ∩ P)-n(B ∩ P)+n(B ∩ C ∩ P)
=52+71+40-15-8-11+2
=131
   Therefore, there are 131 students taking one or
more science courses & (140-131) 9 students
not taking science in Gr 12
4.3 Probability Using Set
Notation
   Venn Diagram
work outward.

B                           C
28       13
50
2
9         6

23

P
4.3 Probability Using Set
Notation
   Additive Principle for Unions of 2 Sets
   n(A U B)=n(A) + n (B) – n(A ∩ B)
   Total number of elements in both sets then subtract
the number that have been counted twice
4.3 Probability Using Set
Notation
Probability of the Union of Two Events
   P(A U B) = P(A) + P(B) – P(A ∩ B)
   If A and B have no outcomes in common
(are disjoint), A and B are said to be
mutually exclusive events and therefore –
P(A U B) = P(A) + P(B) {since P(A ∩ B)=0}
4.3 Probability Using Set
Notation
4.3 Probability Using Set
Notation
Extend our SFBSS science student to
probability.
   Determine the probability that a student
selected at random from the group is
taking Biology.

n( B) 52 13
P( B)               37 %
n( S ) 140 35
4.3 Probability Using Set
Notation
   Determine the probability that a student
selected at random from the group is
taking Chemistry or Physics.
P (C  P )  P (C )  P ( P )  P (C  P )
71 40           8
             
140 140 140
103

140
 73.6%
4.3 Probability Using Set
Notation
   Example:
Consider the sets A={5,6,7,8} and
B={3,4,7,10}

What is A ∩ B?
What is A U B?
Make set C that is disjoint to B.
4.3 Probability Using Set
Notation
 Example
Create a Venn diagram for the following
scenerio. 1500 students attend a local high
school. The first school dance had 740
students in attendance, while only 440
attended the second dance. If 285 students
attended both, how many did not go to either
dance? (find n(A U B) and then its
compliment)
4.3 Probability Using Set
Notation
Example
The probability that Sarah will get into
university is 2/5 and the probability that
she will get into college is 3/4. If the
probability that she gets into both is
1/5, what is the probability that she will
get into at least one of the schools?
4.3 Probability Using Set
Notation
 Example
the GTA read the Toronto Star and 60%
watch CityTV. She is also told that 90%
either read the Star or watch CityTV. If she
places an ad in the paper and runs a
commercial, what is the probability that an
adult selected at random in the GTA will see