Chapter 10, Problem 10.13AYK

(a)

To determine

To find: probabilities and describe.

Answer to Problem 10.13AYK

If a child of Caucasian descent in Germany is randomly selected, the probability that the child has freckles is 0.253.

Explanation of Solution

Given:

Calculation:

Starting at the left of the tree diagram, a child with red hair falls into one of the four hair colors. The bottom leftmost branch gives the probability of a red hair child, so $P\left(redhair\right)$ is 0.025.

If a child of Caucasian descent in Germany is randomly selected, the probability that the child has red hair is 0.025.

The three segments going out from the hair branch point carry the conditional probabilities of eye color given hair color. For example, the top middle branch is $P\left(blueeye|blackhair\right)$ . Now use the general multiplication rule to find $P\left(blueeye\right)$ . Recall that the general multiplication rule is:

  $\begin{array}{l}P\left(AandB\right)=P\left(A\right)P\left(B|A\right)\\ P\left(blueeye\right)\\ =\left\{\begin{array}{l}P\left(blueeyeandblackhair\right)+P\left(blueeyeandbrownhair\right)\\ +P\left(blueeyeandblondehair\right)+P\left(blueeyeandredhair\right)\end{array}\right\}\\ =\left\{\begin{array}{l}P\left(blueeye|blackhair\right)P\left(blackhair\right)+P\left(blueeye\text{ | }brownhair\right)P\left(brownhair\right)\\ +P\left(blueeye\text{ | }blondehair\right)P\left(blondehair\right)+P\left(blueeye\text{ | }redhair\right)P\left(redhair\right)\end{array}\right\}\\ =0.030\cdot 0.061+0.259\cdot 0.461+0.562\cdot 0.453+0.473\cdot 0.025\\ =0.388\end{array}$

If a child of Caucasian descent in Germany is randomly selected, the probability that the child has blue eyes is 0.388.

The two segments going out from the eye color branch point carry the conditional probabilities of freckles given eye color when given hair color. For example, the top rightmost branch is $P\left(freckles\left|blueeye\right|blackhair\right)$ . Now use the general multiplication rule for three events to find $P\left(freckles\right)$ . Recall that the general multiplication rule for three events is:

If a child of Caucasian descent in Germany is randomly selected, the probability that the child has freckles is 0.253.

Conclusion:

Therefore, if a child of Caucasian descent in Germany is randomly selected, the probability that the child has freckles is 0.253.

(b)

To determine

To find: probabilities and describe.

Answer to Problem 10.13AYK

If a red hair child of Caucasian descent in Germany is randomly selected, the probability that the child has freckles is 0.812.

Explanation of Solution

Calculation:

Use the general multiplication rule to find $P\left(frecklesandredhair\right)$ . Recall that the general multiplication rule for three events is:

  $\begin{array}{l}P\left(AandBandC\right)=P\left(A\right)P\left(B|A\right)P\left(C|AandB\right)\cdot P\left(frecklesandredhair\right)\\ =\left\{\begin{array}{l}P\left(frecklesandblueeyeandredhair\right)+P\left(frecklesandgreeneyeandredhair\right)\\ +P\left(frecklesandbrowneyeandredhair\right)\end{array}\right\}\\ =\left\{\begin{array}{l}P\left(frecklesandblueeyeandredhair\right)+P\left(frecklesandgreeneyeandredhair\right)\\ +P\left(frecklesandbrowneyeandredhair\right)\end{array}\right\}\\ =\left(0.025\right)\left(0.473\right)\left(0.857\right)+\left(0.025\right)\left(0.405\right)\left(0.767\right)+\left(0.025\right)\left(0.122\right)\left(0.778\right)\\ =0.0203\end{array}$

If a child of Caucasian descent in Germany is randomly selected, the probability that the child has freckles and red hair is 0.0203.

From part (a), $P\left(redhair\right)$ is 0.025. The condition probability of freckles given red hair is

  $\begin{array}{l}P\left(freckles\text{ | }redhair\right)\\ =\frac{P\left(frecklesandredhair\right)}{P\left(redhair\right)}\\ =\frac{0.0203}{0.025}\\ =0.812\end{array}$

If a red hair child of Caucasian descent in Germany is randomly selected, the probability that the child has freckles is 0.812.

Conclusion:

Therefore,if a red hair child of Caucasian descent in Germany is randomly selected, the probability that the child has freckles is 0.812.