Section 7-5Independent Events

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?

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

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

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

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

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?

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

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

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.

Independent Events

Independent Events

Events A and B are independent IFF P (AB )= P (A)iP (B )

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?

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

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

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

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

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

Dependent Events

Dependent Events

Events A and B are dependent when

P (AB )≠ P (A)iP (B )

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?

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

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

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

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

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.

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?

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

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 )

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

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

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.

Homework

Homework

p. 454 #1-21