More Probability and Odds
Math Analysis & Discrete Math – Section 7.2
I. Warm-Up Problems
1.
2.
Two fair dice are thrown.
a. List the outcomes in the sample space.
b.
Determine the probability that the sum of the dice is less than 5.
c.
Determine the probability that the sum of the dice is greater than or equal to 5.
d.
Determine the probability of rolling a sum of 5 or a sum of 10.
e.
Determine the probability that one die shows an even numbered face and the other die shows an odd
numbered face.
Carol estimates that her probability of passing math is 0.8 and her probability of passing English is 0.7. She
estimates that her probability of passing at least one of them is 0.95. What is the probability she will pass both?
[Hint: it may help to draw a Venn Diagram and/or use the analysis we did very early on.]
II. Extending Some Ideas from Set Theory to Probability
Definition
Mutually exclusive events are events that have no common outcomes. So,
E, F mutually exclusive events
⇔ E∩F = ∅
⇒
P(E∪F) = P(E) + P(F)
Theorem: Inclusion/Exclusion Law for Probabilities
For any two events A and B of a sample space S,
P(A # B ) = P(A)+ P(B )" P(A ! B )
Theorem: Probabilities of Complements
Let E be an event in the sample space S. Then
()
P E = 1 ! P(E )
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III. Practice Problems
For each of these problems, draw a Venn Diagram and find the requested probabilities.
Problem 1
Given: A and B are mutually exclusive events such that P(A) = 0.15 and P(B) = 0.34.
Find:
a. P A
( )
()
b.
PB
c.
P( A " B)
d.
P( A " B)
e.
P A "B
f.
P A "B
!
!
!
(
)
(
)
!
!
!Problem 2
Given: E and F such that P(E) = 0.75, P(F) = 0.40, and P(E∩F) = 0.24.
Find:
a. P E
()
( )
b.
PF
c.
P( E " F )
d.
P E "F
e.
P E "F
f.
P E "F
g.
P E "F
!
!
!
(
)
(
)
(
)
(
)
!
!
!
Homework: 7.2: #1-6, 8-18 even, 19-25 odd, 32-37
!
IV. Warm-Up Problem for Odds
In a particularly easy lottery game, the odds of winning are advertised as 3 to 1. What is the probability you win some prize?
What is the probability you go home empty-handed?
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V. What are the Odds?
We sometimes discuss the likelihoods of events in terms of odds. We define odds as follows:
• The odds for an event E are
P(E) to P E
()
•
The odds against an event E are
()
P E to P(E)
We like integers though, so we work with the fractions to get whole numbers.
!
Finding Odds
!
a
If the odds for an event E are a to b, then
P(E) =
a +b
b
If the odds against an event E are a to b, then
P(E) =
a +b
!
Example
The probability of event E, that it will rain, is 0.3.
a. What are the odds for rain? !
b.
What are the odds against rain?
Example
The odds against the Chicago Cubs willing the league pennant are 200 to 1. What is the probability that the Cubs win the
pennant?
VI. Practice with Probability and Odds
1.
Determine the odds for and against each event from the given probability.
a. P(E) = 0.4
odds for E:
b.
P(E) =
3
8
odds for E:
2.
odds against E:
odds against E:
Determine the probability of E for the given odds.
a. 4 to 1 for E
b.
5 to 4 against E
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3.
The probability that it will rain on Saturday is 0.7. What are the odds against rain on Saturday?
4.
A single die is rolled once, then a fair coin is tossed once. Determine the odds for rolling a number greater than 4,
then tossing a head.
5.
It has been estimated that in 70% of the fatal accidents involving two cars, at least one of the drivers is drunk. If you
hear of a two-car fatal accident, what odds should you give a friend for the event that at least one of the drivers was
drunk?
Homework: 7.2: #41-55 odd
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