Basic Practice of Statistics (Instructor's)
Basic Practice of Statistics (Instructor's)
8th Edition
ISBN: 9781319057923
Author: Moore
Publisher: MAC HIGHER
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Chapter 13, Problem 13.40E

(a)

To determine

To find: The probability of red hair.

To obtain: The probability of blue eyes.

To evaluate: The probability of freckles.

To describe: The obtained probabilities in plain English.

(a)

Expert Solution
Check Mark

Answer to Problem 13.40E

The probability of red hair is 0.0251.

The probability of blue eyes is 0.3876.

The probability of freckles is 0.00011.

There are 2.51% of children are with red hair.

Among the four different hair colors and three different eye colors of Caucasian child, there are 38.76% chances that a child has blue eyes.

Among the four different hair colors and three different eye colors of Caucasian child, there are 0.011% chances that a child has freckles.

Explanation of Solution

Given info:

The tree diagram shows the effect of eye color, hair color and freckles on the reported extent of burning from sun. The last two braches represent “Freckles and No Freckles” categories for each category.

Basic Practice of Statistics (Instructor's), Chapter 13, Problem 13.40E

Calculation:

The probability of red hair is obtained below:

P(redhair)=[(0.025×0.473×0.857+0.025×0.473×0.143)+(0.025×0.405×0.767+0.025×0.405×0.233)+(0.025×0.122×0.778+0.025×0.122×0.222)]=[(0.0101+0.0017)+(0.0078+0.0024)+(0.0024+0.0007)]=(0.0118+0.0102+0.0031)=0.0251

Thus, the probability of red hair is 0.0251.

Interpretation:

Among the four different hair colors (black, brown, blonde, and red) of Caucasian child, there are 2.51% chances that a child has red hair.

Probability of blue eyes:

From the tree diagram,

P(blue eyes | black hair)=0.030P(blue eyes | brown hair)=0.259P(blue eyes | blonde hair)=0.562P(blue eyes | red hair)=0.473

Hence,

P(blue eyes)={P(blue eyes | black hair)P(black hair)+P(blue eyes | brown hair)P(brown hair)+P(blue eyes | blonde hair)P(blonde hair)+P(blue eyes | red hair)P(red hair)}=(0.030)(0.061)+(0.259)(0.461)+(0.562)(0.453)+(0.473)(0.025)=0.00183+0.119399+0.254586+0.011825=0.3876

Thus, the probability of blue eyes is 0.3876.

Interpretation:

Among the four different hair colors (black, brown, blonde, and red) and three different eye colors (blue, green, and brown) of Caucasian child, there are 38.77% chances that a child has blue eyes.

The probability of freckles is obtained below:

From the tree diagram,

P(freckles)={{[P(freckles | black hair and blue eyes)P(black hair and blue eyes)]×[P(freckles | black hair and green eyes)P(black hair and green eyes)]×[P(freckles | black hair and brown eyes)P(black hair and brown  eyes)]}+{[P(freckles | brown hair and blue eyes)P(brown hair and blue eyes)]×[P(freckles | brown hair and green eyes)P(brownhair and green eyes)]×[P(freckles | brown hair and brown eyes)P(brown hair and brown  eyes)]}+{[P(freckles | blonde hair and blue eyes)P(blonde hair and blue eyes)]×[P(freckles | blonde hair and green eyes)P(blondehair and green eyes)]×[P(freckles | blonde hair and brown eyes)P(blonde hair and brown  eyes)]}+{[P(freckles | red hair and blue eyes)P(red hair and blue eyes)]×[P(freckles | red hair and green eyes)P(redhair and green eyes)]×[P(freckles | red hair and brown eyes)P(red hair and brown  eyes)]}}={{(0.143)(0.0018)(0.182)(0.0029)(0.018)(0.0563)}+{(0.278)(0.1194)(0.232)(0.1507)(0.153)(0.1909)}+{(0.319)(0.2546)(0.302)(0.1427)(0.164)(0.0557)}+{(0.857)(0.0118)(0.767)(0.0101)(0.778)(0.0031)}}={0.00005+0.00003+0.00003+0.00000002}=0.00011

Thus, the probability of freckles is 0.00011.

Interpretation:

Among the four different hair colors (black, brown, blonde, and red) and three different eye colors (blue, green, and brown) of Caucasian child, there are 12.48% chances that a child has freckles.

(b)

To determine

To find: The probability of freckles and red hair.

To evaluate: The conditional probability of freckles given that red hair.

(b)

Expert Solution
Check Mark

Answer to Problem 13.40E

The probability of freckles and red hair is 0.0203.

The probability of freckles given that hair is red is 0.8088.

Explanation of Solution

Calculation:

From the Tree diagram of children of Caucasian descent, the required probability is,

P(frecklesandredhair)={[P(red hair)P(blue eyes | red hair)P(freckles | red hair and blue eyes)]+[P(red hair)P(green eyes | red hair)P(freckles | red hair and green eyes)]+[P(red hair)P(brown eyes | red hair)P(freckles | red hair and brown eyes)]}=(0.025×0.473×0.857+0.025×0.405×0.767+0.025×0.122×0.778)=(0.0101+0.0078+0.0024)=0.0203

Thus, the probability of freckles and red hair is 0.0203.

Interpretation:

Among the freckles and no freckles with red hair of Caucasian child, there are 2.03% chances that a child has freckles and red hair.

The conditional probability of freckles given that red hair:

From the tree diagram of children of Caucasian descent, the required probability is,

P(freckles|redhair)=P(frecklesandredhair)P(redhair)=(0.025×0.473×0.857+0.025×0.405×0.767+0.025×0.122×0.778)0.0251=(0.0101+0.0078+0.0024)0.0251=0.02030.0251=0.8088

Thus, the conditional probability of freckles given that hair is red is 0.8088.

Interpretation:

Among the freckles with red hair of Caucasian child, there are 80.88% chances that a child has freckles given red hair.

(c)

To determine

To check: Whether the events “freckles” and “red hair” are independent events.

(c)

Expert Solution
Check Mark

Answer to Problem 13.40E

The events “freckles” and “red hair” are not independent.

Explanation of Solution

The probability of freckles is 0.00011 and the probability of red hair is 0.0251. Also, the probability of freckles and red hair is 0.0203.

P(red hair)P(freckles)=0.025×0.00011=0.00000275

Hence,

P(freckles and red hair)P(freckles)P(red hair)

Thus, the events “freckles” and “red hair” are not independent.

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