MATH 166 Spring 2016
c
Wen
Liu
1.5
1.5 Rules for Probability
Rules:
• For any event E in a sample space S we have
0 ≤ P (E) ≤ 1,
P (S) = 1,
P (∅) = 0
• For two events E and F , we have
P (E ∪ F ) = P (E) + P (F ) − P (E ∩ F )
If E and F are mutually exclusive events, that is, E ∩ F = ∅, then
P (E ∪ F ) = P (E) + P (F )
• P (E C ) = 1 − P (E)
P (E) = 1 − P (E C )
Examples:
1. A single card is randomly drawn from a standard deck of cards. What is the probability that
it will be a red card or a king?
2. Let S = {s1 , s2 , s3 , s4 , s5 , s6 } be the sample space associated with an experiment having the
following partial probability distribution.
Consider the events:
A = {s1 , s2 , s5 }
B = {s3 , s5 , s6 }
C = {s1 , s3 , s4 , s6 }
D = {s1 , s2 , s3 }
Calculate the following probabilities.
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MATH 166 Spring 2016
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Liu
(a) P (s2 )
(b) P (C)
(c) P (B C )
(d) P (AC ∩ B)
(e) P (C C ∪ D)
3. Let E and F be two events of an experiment with sample space S. Suppose P (E) = 0.56, P (F ) =
0.36, and P (E C ∩ F ) = 0.2. Calculate the probabilities P (E C ) and P (E C ∩ F C ).
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MATH 166 Spring 2016
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4. A company surveyed 1000 people on their age and the number of jeans purchased annually.
The results of the poll are shown in the table. A person is selected at random. Compute the
probability of the following.
(a) The person is over 25 and purchases 3 or more pairs of jeans annually.
(b) The person is older than 18 or purchases 1 pair of jeans annually.
(c) The person is in the age group 12-18 and purchases at most 2 pairs of jeans annually.
5. A poll was conducted among 250 residents of a certain city regarding tougher gun-control laws.
The results of the poll are shown in the table.
(a) If one of the participants is selected at random, what is the probability that he or she
favors tougher gun-control laws?
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MATH 166 Spring 2016
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c
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Liu
(b) If one of the participants is selected at random, what is the probability that he or she
owns a handgun?
(c) If one of the participants is selected at random, what is the probability that he or she
owns a handgun but not a rifle?
(d) If one of the participants is selected at random, what is the probability that he or she
favors tougher gun-control laws and does not own a handgun?
Odds
Definition: The odds in favor of an event E are defined to be the ratio of P (E) to P (E C ), or
P (E)
P (E)
=
C
P (E )
1 − P (E)
Often the ratio is reduced to lowest terms, a/b, and then we say that the odds are a to b or a : b.
If the odds for an event E are given as a/b, we can calculate the probability P (E) as follows:
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MATH 166 Spring 2016
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Wen
Liu
1.5
Examples:
1. At the race track, the odds for a horse winning is listed 3 to 2. What is the probability that
the horse will win?
2. A salesman makes two stops when in Pittsburgh. The first stop yields a sale 10% of the time,
the second stop 15% of the time, and both stops yield a sale 4% of the time. Find the odds
that
(a) he will make a sale at both stops. Reduced to lowest terms, odds are
to
.
(b) he will make a sale only at the first stop. Reduced to lowest terms, odds are
(c) he will make a sale at the second stop. Reduced to lowest terms, odds are
to
to
.
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