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In a distant land, parents continue to have children until they have a girl and then they stop having kids. Assume there is no limit to the number of births possible to each couple.

(a) What is the expected number of girls in each family?

(b) What is the expected number of boys in each family?

Part a is 1 since they stop as soon as they have a girl. What is part b? I am trying to apply geometric series but I can't find a solution.

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Two approaches:

  • Presumably half of all births are boys and half of all births are girls, so there are equal numbers of boys and girls across the population (in real life, slightly more boys than girls are born)

  • There is a probability of $\frac12$ of $0$ boys in a family, $\frac14$ of $1$, $\frac18$ of $2$ etc. so you want $\displaystyle\sum_{i=0}^\infty i/2^{i+1}$. If you do not know how to calculate this, try differentiating both sides of $\displaystyle\sum_{i=0}^\infty x^{-i} = x/(x-1)$.

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1 girl & 0 boy is 50%, because it's 50:50 with first kid.

1 girl & 1 boys is 25%, again it's 50:50 of the remaining 50%.

1 girl & n boys is $0.5^{n+1}$ etc..

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The expected number of boys is the mean of the geometric distribution with $p = 0.5$ (which equals $\frac {1} {0.5} = 2$), minus 1 (as we discount the girl). which means 1.

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  • $\begingroup$ So you are saying that the expected number of boys in each family is 1. What about the family where the first child is a girl??? $\endgroup$ – user98280 Oct 2 '13 at 20:57
  • $\begingroup$ @user98280 The meaning of "expected" is basically what the average family will have, obviously some families will have 0 boys and some may have even 20 boys... that's probability. $\endgroup$ – Ron Teller Oct 2 '13 at 21:10

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