Question

1. For every family with 2 children, denote the sex of the first and the second child by a pair xy (where x and y can be either a boy or a girl), and suppose all 4 possible combinations of xy are equally likely.(a) What is the conditional probability that the second child is a girl, given that the first child is a boy. (b) The king comes from a family of two children. What is the probability that his sibling is his sister?

Answer 1

a)as events are independent therefore

P(second child is girl|first child is boy)=P(second child is girl)=1/2

b)

P(one child is boy (King)) =P(both child are boys)+P(one child is boy and other girl)

=P(x=boy and y=boy)+P(x=boy and y=girl)+P(x=girl and y=boy)

=(1/2)*(1/2)+(1/2)*(1/2)+(1/2)*(1/2) =3/4

hence P(other child is a girl(sister) given one child is boy(king))

=P(one child is boy and other girl)/P(one child is boy)

=(P(x=boy and y=girl)+P(x=girl and y=boy))/P(one child is boy)

=((1/2)*(1/2)+(1/2)*(1/2))/(3/4)=(1/2)/(3/4)=2/3