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.

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.

Calculation:

The probability of red hair is obtained below:

$\begin{array}{c}P\left(\text{red}\text{hair}\right)=\left[\begin{array}{l}\left(\begin{array}{l}0.025\times 0.473\times 0.857+\\ 0.025\times 0.473\times 0.143\end{array}\right)+\\ \left(\begin{array}{l}0.025\times 0.405\times 0.767+\\ 0.025\times 0.405\times 0.233\end{array}\right)+\\ \left(\begin{array}{l}0.025\times 0.122\times 0.778+\\ 0.025\times 0.122\times 0.222\end{array}\right)\end{array}\right]\\ =\left[\begin{array}{l}\left(0.0101+0.0017\right)+\\ \left(0.0078+0.0024\right)+\\ \left(0.0024+0.0007\right)\end{array}\right]\\ =\left(0.0118+0.0102+0.0031\right)\\ =0.0251\end{array}$

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,

$\begin{array}{c}P\left(\text{blue eyes | black hair}\right)=0.030\\ P\left(\text{blue eyes | brown hair}\right)=0.259\\ P\left(\text{blue eyes | blonde hair}\right)=0.562\\ P\left(\text{blue eyes | red hair}\right)=0.473\end{array}$

Hence,

$\begin{array}{c}P\left(\text{blue eyes}\right)=\left\{\begin{array}{l}P\left(\text{blue eyes | black hair}\right)P\left(\text{black hair}\right)\\ +P\left(\text{blue eyes | brown hair}\right)P\left(\text{brown hair}\right)\\ +P\left(\text{blue eyes | blonde hair}\right)P\left(\text{blonde hair}\right)\\ +P\left(\text{blue eyes | red hair}\right)P\left(\text{red hair}\right)\end{array}\right\}\\ =\left(0.030\right)\left(0.061\right)+\left(0.259\right)\left(0.461\right)+\left(0.562\right)\left(0.453\right)+\left(0.473\right)\left(0.025\right)\\ =\text{0}\text{.00183}+\text{0}\text{.119399}+\text{0}\text{.254586}+\text{0}\text{.011825}\\ =0.3876\end{array}$

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,

$\begin{array}{c}P\left(\text{freckles}\right)=\left\{\begin{array}{c}\left\{\begin{array}{l}\left[\begin{array}{l}P\left(\text{freckles | black hair and blue eyes}\right)\\ P\left(\text{black hair and blue eyes}\right)\end{array}\right]\\ \times \left[\begin{array}{l}P\left(\text{freckles | black hair and green eyes}\right)\\ P\left(\text{black hair and green eyes}\right)\end{array}\right]\\ \times \left[\begin{array}{l}P\left(\text{freckles | black hair and brown eyes}\right)\\ P\left(\text{black hair and brown eyes}\right)\end{array}\right]\end{array}\right\}\\ +\left\{\begin{array}{l}\left[\begin{array}{l}P\left(\text{freckles | brown hair and blue eyes}\right)\\ P\left(\text{brown hair and blue eyes}\right)\end{array}\right]\\ \times \left[\begin{array}{l}P\left(\text{freckles | brown hair and green eyes}\right)\\ P\left(\text{brown}\text{hair and green eyes}\right)\end{array}\right]\\ \times \left[\begin{array}{l}P\left(\text{freckles | brown hair and brown eyes}\right)\\ P\left(\text{brown hair and brown eyes}\right)\end{array}\right]\end{array}\right\}\\ +\left\{\begin{array}{l}\left[\begin{array}{l}P\left(\text{freckles | blonde hair and blue eyes}\right)\\ P\left(\text{blonde hair and blue eyes}\right)\end{array}\right]\\ \times \left[\begin{array}{l}P\left(\text{freckles | blonde hair and green eyes}\right)\\ P\left(\text{blonde}\text{hair and green eyes}\right)\end{array}\right]\\ \times \left[\begin{array}{l}P\left(\text{freckles | blonde hair and brown eyes}\right)\\ P\left(\text{blonde hair and brown eyes}\right)\end{array}\right]\end{array}\right\}\\ +\left\{\begin{array}{l}\left[\begin{array}{l}P\left(\text{freckles | red hair and blue eyes}\right)\\ P\left(\text{red hair and blue eyes}\right)\end{array}\right]\\ \times \left[\begin{array}{l}P\left(\text{freckles | red hair and green eyes}\right)\\ P\left(\text{red}\text{hair and green eyes}\right)\end{array}\right]\\ \times \left[\begin{array}{l}P\left(\text{freckles | red hair and brown eyes}\right)\\ P\left(\text{red hair and brown eyes}\right)\end{array}\right]\end{array}\right\}\end{array}\right\}\\ =\left\{\begin{array}{c}\left\{\left(0.143\right)\left(0.0018\right)\left(0.182\right)\left(0.0029\right)\left(0.018\right)\left(0.0563\right)\right\}\\ +\left\{\left(0.278\right)\left(0.1194\right)\left(0.232\right)\left(0.1507\right)\left(0.153\right)\left(0.1909\right)\right\}\\ +\left\{\left(0.319\right)\left(0.2546\right)\left(0.302\right)\left(0.1427\right)\left(0.164\right)\left(0.0557\right)\right\}\\ +\left\{\left(0.857\right)\left(0.0118\right)\left(0.767\right)\left(0.0101\right)\left(0.778\right)\left(0.0031\right)\right\}\end{array}\right\}\\ =\left\{0.00005+0.00003+0.00003+0.00000002\right\}\\ =\boxed{0.00011}\end{array}$

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.

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,

$\begin{array}{c}P\left(\text{freckles}\text{and}\text{red}\text{hair}\right)=\left\{\begin{array}{l}\left[\begin{array}{l}P\left(\text{red hair}\right)P\left(\text{blue eyes | red hair}\right)\\ P\left(\text{freckles | red hair and blue eyes}\right)\end{array}\right]\\ +\left[\begin{array}{l}P\left(\text{red hair}\right)P\left(\text{green eyes | red hair}\right)\\ P\left(\text{freckles | red hair and green eyes}\right)\end{array}\right]\\ +\left[\begin{array}{l}P\left(\text{red hair}\right)P\left(\text{brown eyes | red hair}\right)\\ P\left(\text{freckles | red hair and brown eyes}\right)\end{array}\right]\end{array}\right\}\\ =\left(\begin{array}{l}0.025\times 0.473\times 0.857+\\ 0.025\times 0.405\times 0.767+\\ 0.025\times 0.122\times 0.778\end{array}\right)\\ =\left(0.0101+0.0078+0.0024\right)\\ =0.0203\end{array}$

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,

$\begin{array}{c}P\left(\begin{array}{l}\text{freckles|}\\ \text{red}\text{hair}\end{array}\right)=\frac{P\left(\begin{array}{l}\text{freckles}\text{and}\\ \text{red}\text{hair}\end{array}\right)}{P\left(\text{red}\text{hair}\right)}\\ =\frac{\left(\begin{array}{l}0.025\times 0.473\times 0.857+\\ 0.025\times 0.405\times 0.767+\\ 0.025\times 0.122\times 0.778\end{array}\right)}{0.0251}\\ =\frac{\left(0.0101+0.0078+0.0024\right)}{0.0251}\\ =\frac{0.0203}{0.0251}\\ =0.8088\end{array}$

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.

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.

$\begin{array}{c}P\left(\text{red hair}\right)P\left(\text{freckles}\right)=0.025\times 0.00011\\ =\text{0}\text{.00000275}\end{array}$

Hence,

$P\left(\text{freckles and red hair}\right)\ne P\left(\text{freckles}\right)P\left(\text{red hair}\right)$

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