Question

Let A = event that all of a family's children are the same sex and let B = event that the family has at most one boy. Assume P(girl) = P(boy) = 0.5. Test events A and B for independence:

a) if the family has 2 children

b) if the family has 3 children

Answer 1

a)

P(A) =P(all girl)+P(all boy) =(0.5)*(0.5)+(0.5)*(0.5) =0.5

P(B) =P(At most 1 boy) =P(0 boy)+P(1 boy) =(₂C₀)(0.5)⁰(0.5)²+(₂C₁)(0.5)¹(0.5)¹=0.75

P(A n B) =P(at most 1 boy and all of same gender )=P(all girl) =(0.5)*(0.5) =0.25

since P(A)*P(B) =0.5*0.75 =0.375 is not equal to P(A n B) , events are not independent,

b)

P(A) =P(all girl)+P(all boy) =(0.5)*(0.5)*(0.5)+(0.5)*(0.5)*(0.5) =0.25

P(B) =P(At most 1 boy) =P(0 boy)+P(1 boy) =(₃C₀)(0.5)⁰(0.5)³+(₃C₁)(0.5)¹(0.5)²=0.5

P(A n B) =P(at most 1 boy and all of same gender )=P(all girl) =(0.5)*(0.5)*0.5 =0.125

since P(A)*P(B) =0.25*0.5 =0.125 is equal to P(A n B) , events are independent,