Question

Let A be the event that a family has children of both sexes, and let B = the event that a family has at most one boy.

(i) Show that A and B are independent events if a family has three children

(ii) Show that A and B are dependent events if a family has two children.

Answer 1

a)P(A)=P(1 boy and 2 girl)+P(2 boy and 1 girl)=³C₁(0.5)¹*(0.5)²+³C₂(0.5)²*(0.5)¹=0.75

P(B)=P(0 boys )+P(1 boys)=³C₀(0.5)⁰*(0.5)³+³C₁(0.5)¹*(0.5)²=0.5

P(A n B)=P(1 boys)=³C₁(0.5)¹*(0.5)²=0.375

as P(A)*P(B)=P(A n B) ; therefore A and B are independent,

b)

P(A)=P(1 boy and 1 girl)=²C₁(0.5)¹*(0.5)¹ =0.5

P(B)=P(0 boys )+P(1 boys)=²C₀(0.5)¹*(0.5)¹+²C₁(0.5)¹*(0.5)¹=0.75

P(A n B)=P(1 boy and 1 girl)=0.5

as P(A n B) is not equal to P(A)*P(B) ; therefore not independent,