Question

In: Math

Consider an experiment in which a random family is selected among all families with exactly two...

Consider an experiment in which a random family is selected among all families with exactly two children of which one is known to be a girl.

a. Write out the sample space and state the probability that the selected family has two girls.

b. Now consider an experiment in which we select a family randomly among all families with exactly two children, of which one is known to be a girl born on Tuesday. Write out the entire sample space taking into account the gender of the children and the day of the week they were born on.

c. What is the probability that the selected family has two girls?

d. Are the variables “day of the week” and “gender of child” dependent or independent?

Solutions

Expert Solution

Note the order in (a,b) represents a as first child and b as second child.

a. As given, the family selected has two children of which one is girl, so, the other child can be a boy or a girl.

Hence, sample space is (Boy,Girl), (Girl, Boy) and (Girl, Girl).

Probability that family has two girls = 1/3 = 0.33

b. Of the two children, one is girl born in Tuesday.

The other child can either be a girl or a boy. Each gender of the other child can have seven possible days of birth.

Hence, Sample space is as follows -

(Girl-Tuesday, Boy-Sunday), (Girl-Tuesday, Boy-Monday), (Girl-Tuesday, Boy-Tuesday), (Girl-Tuesday, Boy-Wednesday), (Girl-Tuesday, Boy-Thursday), (Girl-Tuesday, Boy-Friday), (Girl-Tuesday, Boy-Saturday), (Girl-Tuesday, Girl-Sunday), (Girl-Tuesday, Girl-Monday), (Girl-Tuesday, Girl-Tuesday), (Girl-Tuesday, Girl-Wednesday), (Girl-Tuesday, Girl-Thursday), (Girl-Tuesday, Girl-Friday), (Girl-Tuesday, Girl-Saturday), (Girl-Sunday, Girl-Tuesday), (Girl-Monday, Girl-Tuesday), (Girl-Wednesday, Girl-Tuesday), (Girl-Thursday, Girl-Tuesday), (Girl-Friday, Girl-Tuesday), (Girl-Saturday, Girl-Tuesday), (Boy-Sunday, Girl-Tuesday), (Boy-Monday, Girl-Tuesday), (Boy-Tuesday, Girl-Tuesday), (Boy-Wednesday, Girl-Tuesday), (Boy-Thursday, Girl-Tuesday), (Boy-Friday, Girl-Tuesday), (Boy-Saturday, Girl-Tuesday)

Sample Size = 27

c. The sample space for two girls with at least one Tues will have 14 combinations but Tues/Tues comes twice which becomes redundant. So, we take only one Tues/Tues combination instead of two.

Selected family having two girls are - (Girl-Tuesday, Girl-Sunday), (Girl-Tuesday, Girl-Monday), (Girl-Tuesday, Girl-Tuesday), (Girl-Tuesday, Girl-Wednesday), (Girl-Tuesday, Girl-Thursday), (Girl-Tuesday, Girl-Friday), (Girl-Tuesday, Girl-Saturday), (Girl-Sunday, Girl-Tuesday), (Girl-Monday, Girl-Tuesday), (Girl-Wednesday, Girl-Tuesday), (Girl-Thursday, Girl-Tuesday), (Girl-Friday, Girl-Tuesday), (Girl-Saturday, Girl-Tuesday)

Favourable outcomes = 13

Probability of two girls = 13/27 = 0.48

d.The variables “day of the week” and “gender of child” are dependent in this case.

milcah answered 7 months ago
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