A family has 3 children each of whom is a boy or a girl with

A family has 3 children, each of whom is a boy or a girl with probability 1/2. Let A = “there is at most 1 girl” B = “the family has children of both sexes.” (a) Are A and B independent? (b) Are A and B independent if the family has 4 children?

Solution

a)

Sample Space = S = { bbb, bbg, bgb,gbb, bgg,gbg, ggb, ggg}

P(A) = P( At most 1 girl) = 4/8 = 0.5

P(B) = P(Both sexes) = 6/8 = 0.75

P(A and B) = 3/8 = P(A) * P(B)

Hence, they are independent

B)

For four children, sample space =

{ BBBB, BBBG, BBGB, BGBB, GBBB, BBGG, BGGB,GGBB, GBGB, BGBG, GBBG, BGGG, GBGG, GGBG, GGGB, GGGG}

P(A) = 5/16

P(B) = 14/16

P(A and B) = 4/16 which is not the same as P(A) * P(B)

Hence the event are not independent

Hope this helps.