Probability with Venn Diagrams and Set Operations by QTq242X

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									Venn Diagrams and Set Operations:
          Tools for Probability
Properties of the Probability of an Event
    Here are some properties of the probability of an
     event that everyone must remember.
    Let E be an event of a sample space S.
1.   If E is the empty set, then P(E)=0.
        For instance, if two dice are tossed, the probability that the sum of
         the faces that turn up is less than 2 is 0.

2.   If E is the whole sample space S, then P(E)=1.
        For instance, if two dice are tossed, the probability that the sum of
         the faces that turn up is between 2 and 12, inclusive, is 1.

3.   Otherwise, 0<P(E)<1.
        That is, the probability of an event is always positive
         and is never more than 1.
   Sets and Venn diagrams can help us investigate
    other interesting properties of the probability of
    an event.
                      Example #1
   12,000 people voted for a politician in his first
    election, 15,000 voted for him in his second,
    and 3,000 voted for him in both elections.
    55,000 people voted in the elections.

   What is the probability that a randomly chosen
    voter voted for the politician
       in at least 1 one of the elections?
       in neither one of the elections?
   We can extend the use of Venn diagrams
    by filling them with probabilities of events
    in a sample space S.

   But now, all the probabilities in the Venn
    diagram must add up to 1.
                  Example #2
    Let A and B be two events of a sample space
     S such that
     p(A)=0.45, p(B)=0.35, and p(A∩B)=0.15.

    Use the given information to determine:
a)   p(AUB)
b)   p(A’∩B)
c)   p(A’∩B’)
                   Example #3
    The manager of a repair shop has observed that
     a car will require a tune-up with a probability of
     0.6, a brake job with a probability of 0.1, and
     both with a probability of 0.02.

    What is the probability that a car will require
a)   either a tune-up or a brake job?
b)   a tune-up but not a brake job?
c)   neither type of repair?
        The Complementary Rule
    If the probability of getting your dream
    job by age 30 is 0.25,

           then the probability of not getting
                it by that age is 0.75

   Thus if E is an event of a sample space S, then

   p(E)=1-p(E’)       These results are referred to
                       as the Complementary Rule
   p(E’)=1-p(E)
                  Example #4
   A bin in a bargain outlet contains 100 blank
    cassette tapes, of which 15 are known to be
    defective.
   If a customer selects 20 of the tapes, determine
    the probability that at least 1 of them is
    defective.
   The Complementary Rule that we have
    encountered is used in a memorable
    mathematics problem:

      The Birthday Problem

								
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