notes 7-5
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Independent EventsTRANSCRIPT
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Section 7-5Independent Events
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Warm-upSuppose the probability that a boy is born in a family is .49 and that
a family has three children of different ages.
1. What is the probability that the oldest two children are boys?
2. What is the probability that the youngest child is a girl?
![Page 3: Notes 7-5](https://reader035.vdocuments.net/reader035/viewer/2022062513/555390dab4c905ba078b4bb6/html5/thumbnails/3.jpg)
Warm-upSuppose the probability that a boy is born in a family is .49 and that
a family has three children of different ages.
1. What is the probability that the oldest two children are boys?
2. What is the probability that the youngest child is a girl?
.49i.49
![Page 4: Notes 7-5](https://reader035.vdocuments.net/reader035/viewer/2022062513/555390dab4c905ba078b4bb6/html5/thumbnails/4.jpg)
Warm-upSuppose the probability that a boy is born in a family is .49 and that
a family has three children of different ages.
1. What is the probability that the oldest two children are boys?
2. What is the probability that the youngest child is a girl?
.49i.49 = .2401
![Page 5: Notes 7-5](https://reader035.vdocuments.net/reader035/viewer/2022062513/555390dab4c905ba078b4bb6/html5/thumbnails/5.jpg)
Warm-upSuppose the probability that a boy is born in a family is .49 and that
a family has three children of different ages.
1. What is the probability that the oldest two children are boys?
2. What is the probability that the youngest child is a girl?
.49i.49 = .2401
1− .49
![Page 6: Notes 7-5](https://reader035.vdocuments.net/reader035/viewer/2022062513/555390dab4c905ba078b4bb6/html5/thumbnails/6.jpg)
Warm-upSuppose the probability that a boy is born in a family is .49 and that
a family has three children of different ages.
1. What is the probability that the oldest two children are boys?
2. What is the probability that the youngest child is a girl?
.49i.49 = .2401
1− .49 = .51
![Page 7: Notes 7-5](https://reader035.vdocuments.net/reader035/viewer/2022062513/555390dab4c905ba078b4bb6/html5/thumbnails/7.jpg)
Warm-upSuppose the probability that a boy is born in a family is .49 and that
a family has three children of different ages.
3. What is the probability that the oldest two children are boys and the youngest child is a girl?
4. Are the events of questions 1 and 2 independent?
![Page 8: Notes 7-5](https://reader035.vdocuments.net/reader035/viewer/2022062513/555390dab4c905ba078b4bb6/html5/thumbnails/8.jpg)
Warm-upSuppose the probability that a boy is born in a family is .49 and that
a family has three children of different ages.
3. What is the probability that the oldest two children are boys and the youngest child is a girl?
4. Are the events of questions 1 and 2 independent?
.49i.49i.51
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Warm-upSuppose the probability that a boy is born in a family is .49 and that
a family has three children of different ages.
3. What is the probability that the oldest two children are boys and the youngest child is a girl?
4. Are the events of questions 1 and 2 independent?
.49i.49i.51 = .122451
![Page 10: Notes 7-5](https://reader035.vdocuments.net/reader035/viewer/2022062513/555390dab4c905ba078b4bb6/html5/thumbnails/10.jpg)
Warm-upSuppose the probability that a boy is born in a family is .49 and that
a family has three children of different ages.
3. What is the probability that the oldest two children are boys and the youngest child is a girl?
4. Are the events of questions 1 and 2 independent?
.49i.49i.51 = .122451
Yes. The gender of the first two has no effect on the third child.
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Independent Events
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Independent Events
Events A and B are independent IFF P (AB )= P (A)iP (B )
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Example 1Imagine that a fair coin is tossed 4 times. Let A = getting all heads and B = getting all tales. Are A and B independent or dependent?
How do you know?
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Example 1Imagine that a fair coin is tossed 4 times. Let A = getting all heads and B = getting all tales. Are A and B independent or dependent?
How do you know?
Dependent
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Example 1Imagine that a fair coin is tossed 4 times. Let A = getting all heads and B = getting all tales. Are A and B independent or dependent?
How do you know?
Dependent
P (A)= 1
16
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Example 1Imagine that a fair coin is tossed 4 times. Let A = getting all heads and B = getting all tales. Are A and B independent or dependent?
How do you know?
Dependent
P (A)= 1
16 P (B )= 1
16
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Example 1Imagine that a fair coin is tossed 4 times. Let A = getting all heads and B = getting all tales. Are A and B independent or dependent?
How do you know?
Dependent
P (A)= 1
16 P (B )= 1
16
P (AB )= 0
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Example 1Imagine that a fair coin is tossed 4 times. Let A = getting all heads and B = getting all tales. Are A and B independent or dependent?
How do you know?
Dependent
P (A)= 1
16 P (B )= 1
16
P (AB )= 0
P (AB )≠ P (A)iP (B )= 1
256
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Dependent Events
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Dependent Events
Events A and B are dependent when
P (AB )≠ P (A)iP (B )
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Example 2Two six-sided dice are rolled. Let S = the sum is 7 and F = the first
die rolled is 5. Are S and F dependent or independent?
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Example 2Two six-sided dice are rolled. Let S = the sum is 7 and F = the first
die rolled is 5. Are S and F dependent or independent?
P (S )= 6
36= 1
6
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Example 2Two six-sided dice are rolled. Let S = the sum is 7 and F = the first
die rolled is 5. Are S and F dependent or independent?
P (S )= 6
36= 1
6 P (F )= 1
6
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Example 2Two six-sided dice are rolled. Let S = the sum is 7 and F = the first
die rolled is 5. Are S and F dependent or independent?
P (S )= 6
36= 1
6 P (F )= 1
6
P (S F )= 1
36
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Example 2Two six-sided dice are rolled. Let S = the sum is 7 and F = the first
die rolled is 5. Are S and F dependent or independent?
P (S )= 6
36= 1
6 P (F )= 1
6
P (S F )= 1
36
P (S )iP (F )= 1
6i
16= 1
36
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Example 2Two six-sided dice are rolled. Let S = the sum is 7 and F = the first
die rolled is 5. Are S and F dependent or independent?
P (S )= 6
36= 1
6 P (F )= 1
6
P (S F )= 1
36
P (S )iP (F )= 1
6i
16= 1
36
These events are independent.
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Example 3A coach keeps records on his basketball team. When a team member shoots two free throws, 65% of the time the first free throw is made
and 70% of the second free throw is made. What percent of the time to both free throws need to be made in order for these events to be
independent?
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Example 3A coach keeps records on his basketball team. When a team member shoots two free throws, 65% of the time the first free throw is made
and 70% of the second free throw is made. What percent of the time to both free throws need to be made in order for these events to be
independent?Let F = first throw and S = second throw
![Page 29: Notes 7-5](https://reader035.vdocuments.net/reader035/viewer/2022062513/555390dab4c905ba078b4bb6/html5/thumbnails/29.jpg)
Example 3A coach keeps records on his basketball team. When a team member shoots two free throws, 65% of the time the first free throw is made
and 70% of the second free throw is made. What percent of the time to both free throws need to be made in order for these events to be
independent?Let F = first throw and S = second throw
P (F )iP (S )
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Example 3A coach keeps records on his basketball team. When a team member shoots two free throws, 65% of the time the first free throw is made
and 70% of the second free throw is made. What percent of the time to both free throws need to be made in order for these events to be
independent?Let F = first throw and S = second throw
P (F )iP (S ) = .65i.7
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Example 3A coach keeps records on his basketball team. When a team member shoots two free throws, 65% of the time the first free throw is made
and 70% of the second free throw is made. What percent of the time to both free throws need to be made in order for these events to be
independent?Let F = first throw and S = second throw
P (F )iP (S ) = .65i.7 = .455
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Example 3A coach keeps records on his basketball team. When a team member shoots two free throws, 65% of the time the first free throw is made
and 70% of the second free throw is made. What percent of the time to both free throws need to be made in order for these events to be
independent?Let F = first throw and S = second throw
P (F )iP (S ) = .65i.7 = .455
Both free throws need to be me 45.5% of the time to have independent events.
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Homework
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Homework
p. 454 #1-21