3/12/2013
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March 12, 2013 Applied Discrete Mathematics
Week 7: Probability Theory
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Now it’s time to look at
Discrete Probability
March 12, 2013 Applied Discrete Mathematics
Week 7: Probability Theory
2
Discrete Probability
Everything you have learned about counting
constitutes the basis for computing the probability of
events to happen.
In the following, we will use the notion experiment
for a procedure that yields one of a given set of
possible outcomes.
This set of possible outcomes is called the sample
space of the experiment.
An event is a subset of the sample space.
March 12, 2013 Applied Discrete Mathematics
Week 7: Probability Theory
3
Discrete Probability
If all outcomes in the sample space are equally likely,
the following definition of probability applies:
The probability of an event E, which is a subset of a
finite sample space S of equally likely outcomes, is
given by p(E) = |E|/|S|.
Probability values range from 0(for an event that will
never happen) to 1(for an event that will always
happen whenever the experiment is carried out).
March 12, 2013 Applied Discrete Mathematics
Week 7: Probability Theory
4
Discrete Probability
Example I:
An urn contains four blue balls and five red balls.
What is the probability that a ball chosen from the urn
is blue?
Solution:
There are nine possible outcomes, and the event
“blue ball is chosen” comprises four of these
outcomes. Therefore, the probability of this event is
4/9 or approximately 44.44%.
March 12, 2013 Applied Discrete Mathematics
Week 7: Probability Theory
5
Discrete Probability
Example II:
What is the probability of winning the lottery 6/49,
that is, picking the correct set of six numbers out of
49?
Solution:
There are C(49, 6) possible outcomes. Only one of
these outcomes will actually make us win the lottery.
p(E) = 1/C(49, 6) = 1/13,983,816
March 12, 2013 Applied Discrete Mathematics
Week 7: Probability Theory
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Complementary Events
Let E be an event in a sample space S. The
probability of an event –E, the complementary
event of E, is given by
p(-E) = 1 – p(E).
This can easily be shown:
p(-E) = (|S| - |E|)/|S| = 1 - |E|/|S| = 1 – p(E).
This rule is useful if it is easier to determine the
probability of the complementary event than the
probability of the event itself.
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