[Math] The expected number of children

Question

I'm thinking about a probability question.

So one couple wants to have one girl and one boy. What's the expected number of children they would have to achieve this? Assume the couple give birth to a boy is 1/2, so the probability to give a birth to girl is 1/2.

Here is my two cents on this,

I considered 4 situations.

• if the first child is a boy and the second is a boy, this happens
  with probability 1/2*(1/2)= 1/4.Then the process is reset and the
  expected number of children increases by 2.

• if the first child is a girl and the second is a girl, this happens
  with probability 1/2*(1/2)= 1/4.Then the process is reset and the
  expected number of children increases by 2.

• if the first child is a girl and the second is a boy, this happens
  with probability 1/4. Then one boy and one girl is obtained after
  having 2 children.

• if the first child is a girl and the second is a boy, this happens
  with probability 1/4. Then one boy and one girl is obtained after
  having 2 children.

Assume the expected number of children the couple will have is E(x).
then we have the following equation,

E(x) = 1/4 ( 2+ E(x)) + 1/4(2+ E(x)) + 1/4*2 + 1/4*2

solve the equation.I get E(x) = 4

Is my logic correct? Thanks.

Answer 1

You could consider the following: GB GGB GGGB

Notice, its geometric distribution starting after the first baby. We know E(X) for geometric is 2.

Hence the answer is 3