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Introduction to Indepencence
Examples
MATH 105: Finite Mathematics
7-5: Independent Events
Prof. Jonathan Duncan
Walla Walla College
Winter Quarter, 2006
Conclusion
Introduction to Indepencence
Outline
1
Introduction to Indepencence
2
Examples
3
Conclusion
Examples
Conclusion
Introduction to Indepencence
Outline
1
Introduction to Indepencence
2
Examples
3
Conclusion
Examples
Conclusion
Introduction to Indepencence
Examples
Conclusion
Conditional Probability
In the last section we saw that knowing something about one event
can effect the probability of another event.
Example
A survey of pizza lovers asked whether the liked thick (C ) or thin
crust and extra cheese (E ) or regular. You select a person at
random. Use the results below to find Pr[C ] and Pr[C |E ].
Thick Crust
Thin Crust
Pr[C ] =
Extra Cheese
24
12
36
40
2
=
60
3
No Extra Cheese
16
8
24
Pr[C |E ] =
24
2
=
36
3
40
20
60
Introduction to Indepencence
Examples
Conclusion
Conditional Probability
In the last section we saw that knowing something about one event
can effect the probability of another event.
Example
A survey of pizza lovers asked whether the liked thick (C ) or thin
crust and extra cheese (E ) or regular. You select a person at
random. Use the results below to find Pr[C ] and Pr[C |E ].
Thick Crust
Thin Crust
Pr[C ] =
Extra Cheese
24
12
36
40
2
=
60
3
No Extra Cheese
16
8
24
Pr[C |E ] =
24
2
=
36
3
40
20
60
Introduction to Indepencence
Examples
Conclusion
Conditional Probability
In the last section we saw that knowing something about one event
can effect the probability of another event.
Example
A survey of pizza lovers asked whether the liked thick (C ) or thin
crust and extra cheese (E ) or regular. You select a person at
random. Use the results below to find Pr[C ] and Pr[C |E ].
Thick Crust
Thin Crust
Pr[C ] =
Extra Cheese
24
12
36
40
2
=
60
3
No Extra Cheese
16
8
24
Pr[C |E ] =
24
2
=
36
3
40
20
60
Introduction to Indepencence
Examples
Conclusion
Conditional Probability
In the last section we saw that knowing something about one event
can effect the probability of another event.
Example
A survey of pizza lovers asked whether the liked thick (C ) or thin
crust and extra cheese (E ) or regular. You select a person at
random. Use the results below to find Pr[C ] and Pr[C |E ].
Thick Crust
Thin Crust
Pr[C ] =
Extra Cheese
24
12
36
40
2
=
60
3
No Extra Cheese
16
8
24
Pr[C |E ] =
24
2
=
36
3
40
20
60
Introduction to Indepencence
Examples
Conclusion
Independent Events
It is not always the case that information about one event changes
the probability of another event.
Independent Events
Events E and F are called independent if the probability of one is
not changed by having information about the outcome of the
other. That is, Pr[E |F ] = Pr[E ]
Tests for Independence
Test for independence using the formula:
Pr[E ∩ F ] = Pr[E ] · Pr[F ]
or, use a Venn Diagram and determine if the ratio of E ∩ F to F is
the same as the ratio of E to S
Introduction to Indepencence
Examples
Conclusion
Independent Events
It is not always the case that information about one event changes
the probability of another event.
Independent Events
Events E and F are called independent if the probability of one is
not changed by having information about the outcome of the
other. That is, Pr[E |F ] = Pr[E ]
Tests for Independence
Test for independence using the formula:
Pr[E ∩ F ] = Pr[E ] · Pr[F ]
or, use a Venn Diagram and determine if the ratio of E ∩ F to F is
the same as the ratio of E to S
Introduction to Indepencence
Examples
Conclusion
Independent Events
It is not always the case that information about one event changes
the probability of another event.
Independent Events
Events E and F are called independent if the probability of one is
not changed by having information about the outcome of the
other. That is, Pr[E |F ] = Pr[E ]
Tests for Independence
Test for independence using the formula:
Pr[E ∩ F ] = Pr[E ] · Pr[F ]
or, use a Venn Diagram and determine if the ratio of E ∩ F to F is
the same as the ratio of E to S
Introduction to Indepencence
Outline
1
Introduction to Indepencence
2
Examples
3
Conclusion
Examples
Conclusion
Introduction to Indepencence
Examples
Conclusion
A Venn Diagram Example
Example
Let A and B be events in a sample space S such that Pr[A] =
8
2
Pr[B] = 16
, and Pr[A ∩ B] = 16
. Are A and B independent?
Pr[A ∩ B] =
Pr[A]·Pr[B] =
4
16 ,
1
2
=
16
8
4 8
1
· =
16 16
8
Independent!
Introduction to Indepencence
Examples
Conclusion
A Venn Diagram Example
Example
Let A and B be events in a sample space S such that Pr[A] =
8
2
Pr[B] = 16
, and Pr[A ∩ B] = 16
. Are A and B independent?
A
Pr[A ∩ B] =
B
2
16
2
16
6
16
Pr[A]·Pr[B] =
4
16 ,
1
2
=
16
8
4 8
1
· =
16 16
8
6
16
Independent!
Introduction to Indepencence
Examples
Conclusion
A Venn Diagram Example
Example
Let A and B be events in a sample space S such that Pr[A] =
8
2
Pr[B] = 16
, and Pr[A ∩ B] = 16
. Are A and B independent?
A
Pr[A ∩ B] =
B
2
16
2
16
6
16
Pr[A]·Pr[B] =
4
16 ,
1
2
=
16
8
4 8
1
· =
16 16
8
6
16
Independent!
Introduction to Indepencence
Examples
Conclusion
A Venn Diagram Example
Example
Let A and B be events in a sample space S such that Pr[A] =
8
2
Pr[B] = 16
, and Pr[A ∩ B] = 16
. Are A and B independent?
A
Pr[A ∩ B] =
B
2
16
2
16
6
16
Pr[A]·Pr[B] =
4
16 ,
1
2
=
16
8
4 8
1
· =
16 16
8
6
16
Independent!
Introduction to Indepencence
Examples
Conclusion
A Venn Diagram Example
Example
Let A and B be events in a sample space S such that Pr[A] =
8
2
Pr[B] = 16
, and Pr[A ∩ B] = 16
. Are A and B independent?
A
Pr[A ∩ B] =
B
2
16
2
16
6
16
Pr[A]·Pr[B] =
4
16 ,
1
2
=
16
8
4 8
1
· =
16 16
8
6
16
Independent!
Introduction to Indepencence
Examples
Conclusion
Independence and Tree Diagrams
Example
A fair coin is tossed twocie and events E and F are defined as:
E:
F:
Heads on the first toss
Tails on the second toss
Are E and F independent? Use a tree diagram to find out.
Example
In a group of seeds, 31 of which should produce violets, the best
germinateion that can be obtained is 60%. If one seed is planted,
what is the probability it will grow a violet? Assume the events are
independent.
Solve both using a tree diagram and a Venn diagram.
Introduction to Indepencence
Examples
Conclusion
Independence and Tree Diagrams
Example
A fair coin is tossed twocie and events E and F are defined as:
E:
F:
Heads on the first toss
Tails on the second toss
Are E and F independent? Use a tree diagram to find out.
Example
In a group of seeds, 31 of which should produce violets, the best
germinateion that can be obtained is 60%. If one seed is planted,
what is the probability it will grow a violet? Assume the events are
independent.
Solve both using a tree diagram and a Venn diagram.
Introduction to Indepencence
Examples
Conclusion
A Used Car Lot
Example
There are 16 cars on a used car lot: 10 compacts and 6 sedans.
Three of the compacts are blue, the rest are red. Five of the sedans
are blue and the rest are red. A car is picked at random. Are the
events of picking a sedan and picking a blue car independent?
8
1
=
16
2
Pr[D] =
Pr[B ∩ D] =
5
1 3
6= ·
16
2 8
Pr[B] =
6
3
=
16
8
These events are not independent.
Introduction to Indepencence
Examples
Conclusion
A Used Car Lot
Example
There are 16 cars on a used car lot: 10 compacts and 6 sedans.
Three of the compacts are blue, the rest are red. Five of the sedans
are blue and the rest are red. A car is picked at random. Are the
events of picking a sedan and picking a blue car independent?
8
1
=
16
2
Pr[D] =
Pr[B ∩ D] =
5
1 3
6= ·
16
2 8
Pr[B] =
6
3
=
16
8
These events are not independent.
Introduction to Indepencence
Examples
Conclusion
A Used Car Lot
Example
There are 16 cars on a used car lot: 10 compacts and 6 sedans.
Three of the compacts are blue, the rest are red. Five of the sedans
are blue and the rest are red. A car is picked at random. Are the
events of picking a sedan and picking a blue car independent?
8
1
=
16
2
Pr[D] =
Pr[B ∩ D] =
5
1 3
6= ·
16
2 8
Pr[B] =
6
3
=
16
8
These events are not independent.
Introduction to Indepencence
Examples
Conclusion
A Used Car Lot
Example
There are 16 cars on a used car lot: 10 compacts and 6 sedans.
Three of the compacts are blue, the rest are red. Five of the sedans
are blue and the rest are red. A car is picked at random. Are the
events of picking a sedan and picking a blue car independent?
8
1
=
16
2
Pr[D] =
Pr[B ∩ D] =
5
1 3
6= ·
16
2 8
Pr[B] =
6
3
=
16
8
These events are not independent.
Introduction to Indepencence
Outline
1
Introduction to Indepencence
2
Examples
3
Conclusion
Examples
Conclusion
Introduction to Indepencence
Examples
Important Concepts
Things to Remember from Section 7-5
1
Events A and B are independent if
Pr[A ∩ B] = Pr[A] · Pr[B]
2
Events A and B are independent if
Pr[A|B] = Pr[A]
Conclusion
Introduction to Indepencence
Examples
Important Concepts
Things to Remember from Section 7-5
1
Events A and B are independent if
Pr[A ∩ B] = Pr[A] · Pr[B]
2
Events A and B are independent if
Pr[A|B] = Pr[A]
Conclusion
Introduction to Indepencence
Examples
Important Concepts
Things to Remember from Section 7-5
1
Events A and B are independent if
Pr[A ∩ B] = Pr[A] · Pr[B]
2
Events A and B are independent if
Pr[A|B] = Pr[A]
Conclusion
Introduction to Indepencence
Examples
Conclusion
Next Time. . .
Next time we will explore conditional probabilities which are not
easily explored using tree diagrams, such as the final example seen
in section 7-4.
For next time
Read section 8-1
Introduction to Indepencence
Examples
Conclusion
Next Time. . .
Next time we will explore conditional probabilities which are not
easily explored using tree diagrams, such as the final example seen
in section 7-4.
For next time
Read section 8-1