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Probability of Combined Events
Probability and Venn Diagrams
Odds
MATH 105: Finite Mathematics
7-2: Properties of Probability
Prof. Jonathan Duncan
Walla Walla College
Winter Quarter, 2006
Conclusion
Probability of Combined Events
Probability and Venn Diagrams
Outline
1
Probability of Combined Events
2
Probability and Venn Diagrams
3
Odds
4
Conclusion
Odds
Conclusion
Probability of Combined Events
Probability and Venn Diagrams
Outline
1
Probability of Combined Events
2
Probability and Venn Diagrams
3
Odds
4
Conclusion
Odds
Conclusion
Probability of Combined Events
Probability and Venn Diagrams
Odds
Probabilities of Events
Last time we looked at assigning probabilities to outcomes in a
sample space.
Example
An urn contains 10 balls: 5 blue, 3 green, and 2 red. You draw a
ball at random and note its color.
1
Find the probability of the event E = ∅.
2
Find the probability of the “simple event” E = {G }.
3
Find the probability of the event E = {Green, Red}.
Conclusion
Probability of Combined Events
Probability and Venn Diagrams
Odds
Probabilities of Events
Last time we looked at assigning probabilities to outcomes in a
sample space.
Example
An urn contains 10 balls: 5 blue, 3 green, and 2 red. You draw a
ball at random and note its color.
S = {R, G , B}
1
Find the probability of the event E = ∅.
2
Find the probability of the “simple event” E = {G }.
3
Find the probability of the event E = {Green, Red}.
Conclusion
Probability of Combined Events
Probability and Venn Diagrams
Odds
Probabilities of Events
Last time we looked at assigning probabilities to outcomes in a
sample space.
Example
An urn contains 10 balls: 5 blue, 3 green, and 2 red. You draw a
ball at random and note its color.
S = {R, G , B}
1
Find the probability of the event E = ∅.
2
Find the probability of the “simple event” E = {G }.
3
Find the probability of the event E = {Green, Red}.
Conclusion
Probability of Combined Events
Probability and Venn Diagrams
Odds
Probabilities of Events
Last time we looked at assigning probabilities to outcomes in a
sample space.
Example
An urn contains 10 balls: 5 blue, 3 green, and 2 red. You draw a
ball at random and note its color.
S = {R, G , B}
1
Find the probability of the event E = ∅.
2
Find the probability of the “simple event” E = {G }.
3
Find the probability of the event E = {Green, Red}.
Conclusion
Probability of Combined Events
Probability and Venn Diagrams
Odds
Probabilities of Events
Last time we looked at assigning probabilities to outcomes in a
sample space.
Example
An urn contains 10 balls: 5 blue, 3 green, and 2 red. You draw a
ball at random and note its color.
S = {R, G , B}
1
Find the probability of the event E = ∅.
2
Find the probability of the “simple event” E = {G }.
3
Find the probability of the event E = {Green, Red}.
Conclusion
Probability of Combined Events
Probability and Venn Diagrams
Odds
Conclusion
Unions of Events
In the last part of the last problem, we could take E = {Green}
and F = {Red} and note that:
Union of Mutually Exclusive Events
Let E and F be mutually exclusive (disjoint) events in a sample
space S. Then,
Pr [E ∪ F ] = Pr [E ] + Pr [F ]
Example
You roll two fair six-sided dice and note the two numbers showing.
With the sets E and F below, find Pr [E ], Pr [F ], and Pr [E ∪ F ].
E = {(x, y ) | x + y is even }
F = {(x, y ) | x + y ≥ 10}
Probability of Combined Events
Probability and Venn Diagrams
Odds
Conclusion
Unions of Events
In the last part of the last problem, we could take E = {Green}
and F = {Red} and note that:
Union of Mutually Exclusive Events
Let E and F be mutually exclusive (disjoint) events in a sample
space S. Then,
Pr [E ∪ F ] = Pr [E ] + Pr [F ]
Example
You roll two fair six-sided dice and note the two numbers showing.
With the sets E and F below, find Pr [E ], Pr [F ], and Pr [E ∪ F ].
E = {(x, y ) | x + y is even }
F = {(x, y ) | x + y ≥ 10}
Probability of Combined Events
Probability and Venn Diagrams
Odds
Conclusion
Unions of Events
In the last part of the last problem, we could take E = {Green}
and F = {Red} and note that:
Union of Mutually Exclusive Events
Let E and F be mutually exclusive (disjoint) events in a sample
space S. Then,
Pr [E ∪ F ] = Pr [E ] + Pr [F ]
Example
You roll two fair six-sided dice and note the two numbers showing.
With the sets E and F below, find Pr [E ], Pr [F ], and Pr [E ∪ F ].
E = {(x, y ) | x + y is even }
F = {(x, y ) | x + y ≥ 10}
Probability of Combined Events
Probability and Venn Diagrams
Odds
Conclusion
General Unions
General Unions of Events
In general, if E and F are not-necessarily mutually exclusive events
in a sample space S, then
Pr [E ∪ F ] = Pr [E ] + Pr [F ] − Pr [E ∩ F ]
Probability of Combined Events
Probability and Venn Diagrams
Odds
Conclusion
General Unions
General Unions of Events
In general, if E and F are not-necessarily mutually exclusive events
in a sample space S, then
Pr [E ∪ F ] = Pr [E ] + Pr [F ] − Pr [E ∩ F ]
Recall that we used Venn Diagrams to help visualize this rule when
it was stated for counting elements of sets. The same tool can be
used for probability.
Probability of Combined Events
Probability and Venn Diagrams
Outline
1
Probability of Combined Events
2
Probability and Venn Diagrams
3
Odds
4
Conclusion
Odds
Conclusion
Probability of Combined Events
Probability and Venn Diagrams
Odds
Conclusion
Using Venn Diagrams
Example
Let A and B be events in a sample space S with Pr [A] = 0.60,
Pr [B] = 0.40, and Pr [A ∩ B] = 0.25. Use Venn Diagrams to find.
1
Pr [A ∪ B]
2
Pr [A]
3
Pr [A ∩ B]
4
Pr [A ∪ B]
Probability of Combined Events
Probability and Venn Diagrams
Outline
1
Probability of Combined Events
2
Probability and Venn Diagrams
3
Odds
4
Conclusion
Odds
Conclusion
Probability of Combined Events
Probability and Venn Diagrams
Odds
Conclusion
From Probability to Odds
Many times probabilities are not expressed as numbers between 0
and 1, but rather as the odds for or against an event happening.
Finding Odds
If E is an event in a sample space S, then
The odds for E are
Pr [E ]
Pr [E ]
The odds against E are
Pr [E ]
Pr [E ]
Example
If the probability of an event E is 0.2, find the odds for and against
the event.
Probability of Combined Events
Probability and Venn Diagrams
Odds
Conclusion
From Probability to Odds
Many times probabilities are not expressed as numbers between 0
and 1, but rather as the odds for or against an event happening.
Finding Odds
If E is an event in a sample space S, then
The odds for E are
Pr [E ]
Pr [E ]
The odds against E are
Pr [E ]
Pr [E ]
Example
If the probability of an event E is 0.2, find the odds for and against
the event.
Probability of Combined Events
Probability and Venn Diagrams
Odds
Conclusion
From Probability to Odds
Many times probabilities are not expressed as numbers between 0
and 1, but rather as the odds for or against an event happening.
Finding Odds
If E is an event in a sample space S, then
The odds for E are
Pr [E ]
Pr [E ]
The odds against E are
Pr [E ]
Pr [E ]
Example
If the probability of an event E is 0.2, find the odds for and against
the event.
Probability of Combined Events
Probability and Venn Diagrams
Odds
Conclusion
From Odds to Probability
You can also convert odds back into probability as shown below.
Finding Probability
If E is an event in a sample space S with the odds for E are “a to
b” then
Pr [E ] =
a
a+b
Pr [F ] =
b
a+b
Example
If the odds for A are 1 to 5 and the odds against B are 3 to 1,
what are the odds for “A or B” assuming that A and B are
mutually exclusive?
Probability of Combined Events
Probability and Venn Diagrams
Odds
Conclusion
From Odds to Probability
You can also convert odds back into probability as shown below.
Finding Probability
If E is an event in a sample space S with the odds for E are “a to
b” then
Pr [E ] =
a
a+b
Pr [F ] =
b
a+b
Example
If the odds for A are 1 to 5 and the odds against B are 3 to 1,
what are the odds for “A or B” assuming that A and B are
mutually exclusive?
Probability of Combined Events
Probability and Venn Diagrams
Odds
Conclusion
From Odds to Probability
You can also convert odds back into probability as shown below.
Finding Probability
If E is an event in a sample space S with the odds for E are “a to
b” then
Pr [E ] =
a
a+b
Pr [F ] =
b
a+b
Example
If the odds for A are 1 to 5 and the odds against B are 3 to 1,
what are the odds for “A or B” assuming that A and B are
mutually exclusive?
Probability of Combined Events
Probability and Venn Diagrams
Outline
1
Probability of Combined Events
2
Probability and Venn Diagrams
3
Odds
4
Conclusion
Odds
Conclusion
Probability of Combined Events
Probability and Venn Diagrams
Odds
Important Concepts
Things to Remember from Section 7-2
1
Events are just Sets, and can be treated as such.
2
Pr [A ∪ B] = Pr [A] + Pr [B] − Pr [A ∩ B]
3
Conversion between probabilities and odds and back.
Conclusion
Probability of Combined Events
Probability and Venn Diagrams
Odds
Important Concepts
Things to Remember from Section 7-2
1
Events are just Sets, and can be treated as such.
2
Pr [A ∪ B] = Pr [A] + Pr [B] − Pr [A ∩ B]
3
Conversion between probabilities and odds and back.
Conclusion
Probability of Combined Events
Probability and Venn Diagrams
Odds
Important Concepts
Things to Remember from Section 7-2
1
Events are just Sets, and can be treated as such.
2
Pr [A ∪ B] = Pr [A] + Pr [B] − Pr [A ∩ B]
3
Conversion between probabilities and odds and back.
Conclusion
Probability of Combined Events
Probability and Venn Diagrams
Odds
Important Concepts
Things to Remember from Section 7-2
1
Events are just Sets, and can be treated as such.
2
Pr [A ∪ B] = Pr [A] + Pr [B] − Pr [A ∩ B]
3
Conversion between probabilities and odds and back.
Conclusion
Probability of Combined Events
Probability and Venn Diagrams
Odds
Conclusion
Next Time. . .
Now that we are hopefully more comfortable with probabilities, it
is time to start using the counting rules we learned in the last
chapter to help compute probabilities for more complicated events.
In the next section we use Combinations and Permutations to
compute probabilities.
For next time
Read Section 7-3 (pp 386-391)
Do Problem Sets 7-3 A,B
Probability of Combined Events
Probability and Venn Diagrams
Odds
Conclusion
Next Time. . .
Now that we are hopefully more comfortable with probabilities, it
is time to start using the counting rules we learned in the last
chapter to help compute probabilities for more complicated events.
In the next section we use Combinations and Permutations to
compute probabilities.
For next time
Read Section 7-3 (pp 386-391)
Do Problem Sets 7-3 A,B