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May 18, 2011
Section 9.3
Probability
Probability of an Event (equally likely outcomes)
If E is an event in a finite, nonempty
sample space S of equally likely outcomes,
then the probability of the event E is
P(E) =
sample space: the set of all possible outcomes
of an experiment
What is the sample space when a red and
green die are rolled simultaneously?
(1, 1)
(1, 2)
(1, 3)
(1, 4)
(1, 5)
(1, 6)
(2, 1)
(2, 2)
(2, 3)
(2, 4)
(2, 5)
(2, 6)
(3, 1)
(3, 2)
(3, 3)
(3, 4)
(3, 5)
(3, 6)
(4, 1)
(4, 2)
(4, 3)
(4, 4)
(4, 5)
(4, 6)
(5, 1)
(5, 2)
(5, 3)
(5, 4)
(5, 5)
(5, 6)
(6, 1)
(6, 2)
(6, 3)
(6, 4)
(6, 5)
(6, 6)
What is the probability that the sum of
the dice totals 9?
What is the probability that the sum of
the dice totals less than 9?
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Probability Function
A probability function is a function P that
assigns to each outcome in a sample
space a unique real number, subject to the
following conditions:
1. 0 ≤ P(O) ≤ 1 for every outcome O;
2. the sum of the probabilities of all
outcomes in S is 1;
3. P(∅) = 0.
Multiplication Principle of Probability
Suppose an event A has probability p1
and an event B has probability p2 under
the assumption that A occurs. Then the
probability that A and B occur is p1⋅p2.
12. P(not red) = 1 − 0.2 = 0.8
18. P(first brown and second is yellow)
0.3 ⋅ 0.2 = 0.06
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23. P(4 aces) =
Ten dimes dated 2000 through 2009
are tossed. Find the probability of
each event.
P(the 2005 dime is a tails) = 1/2
P(heads on exactly 4 dimes)
P(tails on exactly 4 dimes)
P(tails on exactly 6 dimes)
P(heads on at least 2 dimes)
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Find the probability that no two people in a
group of n people share the same birthday.
Let n be 25.
Consider that no two people share the same
birthday.
notice that
and
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A certain test to detect the presence of the
AIDS antibody will test positive 99.7% of
the time if a person has the antibody and
give a false negative 0.3% of the time. It
will give a false positive 1.5% of the time.
As of 2003, P(HIV/AIDS) for the general
population in the U.S. was 0.006. How
effective is this test.
P(true +) = .997
P(have) = .006
P(false −) = .003
P(false +) = .015
P(not) = .994
P(true −) = .985
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Binomial Probability
Where p is the probability of a success,
and q is the probability of a failure (which
is complementary to p, that is q = 1 − p).
Suppose it is known that 1 out of every 350
light bulbs produced by a certain machine is
defective. Find the probability that 2 bulbs
in a sample of 5 will be defective.
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