What is the probability that {a family has two children, given that it has two girls}, that is P(C2 | G2)?

The random experiment consists in the number of children and their gender in families of a given country. Consider the following generic events for k ≥ 0: • Ck = {a family has k child/children} • Bk = {a family has k boy(s)} • Gk = {a family has k girl(s)} A research centre in Demography uses the following model for the number of children in families: P(Ck) =  1/6 if k = 0 or k = 1 1/ 3 × 2 k−2  if k ≥ 2 and assumes that the probabilities to give birth to a boy or a girl are equal (i.e. 1/2 each). (1) In this Question, we consider only families with k children (i.e. the event {Ck} is satisfied). Specify the related sample space Sk and give |Sk|. Justify why P(G2 | Ck) =  k 2 2 k . [4 pts] (2) By noting that {C0}, {C1}, . . . , {C∞} is a partition of the sample space, derive P(G2). [hint: we can use the formula P∞ n=2 n(n − 1)/4 n−2 = 2 × 4 3/3 3 ] [4 pts] (3) What is the probability that {a family has two children, given that it has two girls}, that is P(C2 | G2)? [4 pts] (4) In this Question, we consider only families with 4 children (i.e. the event {C4} is satisfied). With a similar reasoning to Question (1), derive P(B2 ∩ G2 | C4). [4 pts] (5) What is the probability that {a family has two boys, given that it has two girls}, that is P(B2 | G2)? [hint: one can remark that {G2 ∩ B2} = {G2 ∩ B2 ∩ C4} and use Question (4).]

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