[Math] Expected no. of children “at least one boy and at least one girl, with boy older than girl”

conditional-expectationexpectationgeometric-probabilityprobability

A couple decides to keep having children until

  1. Cond1: they have at least one boy and at least one girl,
  2. Cond2: with boy older than girl

and then stop. Assume they never have twins, that the “trials” are independent with probability $1/2$ of a boy, and that they are fertile enough to keep producing children indefinitely. What is the expected number of children?

Note: updated

If we consider just Cond1, answer would be
Let $X$ be the number of children needed, starting with the 2nd child, to obtain one whose gender is not the same as that of the firstborn. Then $X − 1$ is Geom(1/2), so $E(X) = 2$. This does not include the firstborn, so the expected total number of children is $E(X + 1) = E(X) + 1 = 3$.

Best Answer

Hint:

  • You want a boy born and then a girl
  • What is the expected number of children until the first boy is born?
  • Given that the first boy has been born, how many additional children are expected until the next girl is born?
  • Use linearity of expectation.
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