Suppose that 15 percent of families in a community have no children, 20 percent

Question

Suppose that 15 percent of families in a community have no children, 20 percent have one child, 35 percent have two children, and 30 percent have three children. (No families have more than three in this community.) Suppose further that in each family, each child is equally likely to be a boy or a girl and that the genders of children within a family are independent of each other. If family is chosen at random from this community, the B= the number of boys in the family and G=the number of girls in the family.

1) What is the expected number of boys in a family?

2) If a family has no boys, what is the probability that the family has one girl?

3) If a family has one girl, what is the expected number of boys in the family?

4) What is the probability that family has one girl?

Explanation / Answer

1) Expected number of boys = 0.5x expected number of children)

= 0.5x(0 + 0.20x1 + 0.35x2 + 0.30x3)

= 0.9

2) 50% of single child families has no boys and 1 girl

P(family having no boys and 1 girl) = 0.50 x 0.20 = 0.1

3) 50% of single child families has no boys and 1 girl, 50% of two children family has 1 boy and 1 girl, 37.5% of 3 children family has 2 boys and a girl

Expected number of boys in a one girl family = 0 + 0.50x0.35x1 + 0.375x0.3x2 = 0.4

So, if a family has 1 girl, expected number of boys in that family = 0.4

4) P(a family has 1 girl) = 0.5x0.20 + 0.5x0.35 + 0.375x0.3 = 0.3875

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