Question

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. Find whether events ‘A’ and ‘B’ are independent. Also find whether A, B, C are independent or not.

Answer 1

A family of three children can have 

(i) All 3 boys 

(ii) 2 boys + 1 girl 

(iii) 1 boy + 2 girls 

(iv) 3 girls

Event ‘A’ is associated with (iii) & (iv). Hence P(A) = 1/2

Event ‘B’ is associated with (ii) & (iii). Hence P(B) = 3/4

Event ‘C’ is associated with (i) & (ii). Hence P(C) = 1/2

P(A ∩ B) = P(iii) = 3/8 = P(A).P(B). 

Hence A and B are independent of each other P(A ∩ C) = 0 ≠ P(A). P(C). 

Hence A, B, C are not independent.