Practice of Statistics in the Life Sciences
Practice of Statistics in the Life Sciences
4th Edition
ISBN: 9781319013370
Author: Brigitte Baldi, David S. Moore
Publisher: W. H. Freeman
Question
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Chapter 10, Problem 10.13AYK

(a)

To determine

To find: probabilities and describe.

(a)

Expert Solution
Check Mark

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(red hair) 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(blue eye|black hair) . Now use the general multiplication rule to find P(blue eye) . Recall that the general multiplication rule is:

  P(A and B)=P(A)P(B|A)P(blue eye)={P(blue eye and black hair)+P(blue eye and brown hair)+P(blue eye and blonde hair)+P(blue eye and red hair)}={P(blue eye|black hair)P(black hair)+P(blue eye | brown hair)P(brown hair)+P(blue eye | blonde hair)P(blonde hair)+P(blue eye | red hair)P(red hair)}=0.0300.061+0.2590.461+0.5620.453+0.4730.025=0.388

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(freckles | blue eye | black hair) . Now use the general multiplication rule for three events to find P(freckles) . 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.

(b)

Expert Solution
Check Mark

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(freckles and red hair) . Recall that the general multiplication rule for three events is:

  P(A and B and C)=P(A)P(B|A)P(C|Aand B)P(freckles and red hair)={P(freckles and blue eye and red hair)+P(freckles and green eye and red hair)+P(freckles and brown eye and red hair)}={P(freckles and blue eye and red hair)+P(freckles and green eye and red hair)+P(freckles and brown eye and red hair)}=(0.025)(0.473)(0.857)+(0.025)(0.405)(0.767)+(0.025)(0.122)(0.778)=0.0203

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(red hair) is 0.025. The condition probability of freckles given red hair is

  P(freckles | red hair)=P(freckles and red hair)P(red hair)=0.02030.025=0.812

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.

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