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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.


             What does the shaded
             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
Somethings to think about!
   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
Extended Additive Principle
     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
       Start with overlap amongst 3 courses and
        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 Using Additive Principle:
    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)
Answer = 605
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?
Answer=19/20 or 95%
4.3 Probability Using Set
Notation
 Example
An advertiser is told that 70% of all adults in
  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
  both of these ads?
Answer = 0.4 or 40%
4.3 Probability Using Set
Notation
   Home Entertainment
    Pg 228 #1-15

				
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