A family has three children. Event `'A'` is that family has at most one boy, Event `'B'` is that family has at least one boy and one girl, Event `'C'` is that the family has at most one girl. Then

Options

A. A. Events `'A'` and `'B'` are independent
B. B. Events `'A'` and `'B'` are notindependent
C. C. Events `A,B,C` are independent
D. D. Events `A,B,C` are not independent

Answer

A, D

Explanation

`(a,d)` Clearly `P(A)=P(C )`
Event `A` is no boy or exactly one boy in family
`impliesP(A)=((1)/(2))^(3)+^(3)C_(1)((1)/(2))^(3)=(1)/(8)+(3)/(8)=(1)/(2)`
Event `B` is `2` boy, `1` girl or `1` boy, `2` girls
`P(B)=^(3)C_(2)((1)/(2))^(3)+^(3)C_(2)((1)/(2))^(3)=(3)/(4)`
Event `C` is no girl or exactly one girl
`P(C )=((1)/(2))^(3)+^(3)C_(1)((1)/(2))^(3)=(1)/(2)`
Event `AnnB` is one boy and two girls
`P(AnnB)=^(3)C_(1)((1)/(2))^(3)=3//8`
Event `BnnC` is one girl and two boys `P(BnnC)=(3)/(8)`
`AnnCto phi`
`impliesAnnBnnC` in also `phi`
`P(AnnB)=(3)/(8)=P(A)xxP(B)`, so `A` and `B` are independent neither `P(AnnBnnC)=P(A)xxP(B)xxP(B)` nor `P(AnnC)=P(A)xxP(C )`
So `ABC` is not independent.