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In Culture A, parents have children until they have a boy and a girl (not necessarily in that order). In Culture B parents have children until they have 2 boys, regardless of the number of girls. Assuming children are born one at a time, that boys and girls are equally probable, and that families can have as many children as they like until they decide to stop, calculate (a) the expected number of children a Culture A family will have, and (b) the expected number of children a Culture B family will have. (Hint: condition on the sex of the first child born.) (c) What is the expected number of boys and girls in each case?

For part a , I get the expected number is 3. Is it correct? In my thinking that Let X be the number of children. And if the first child is boy or girl, then (X-1)~geom(1/2). Thus, E(X-1)=2. And E(X)=1+E(X-1)=1+2=3.

In part b I got the expected number is 4. So, part a and b is done.

For Part C the question ask boys and girls in each case?Does is mean I need to find 4 expected numbers , 1 for boys and 1 for girls in each case?

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  • $\begingroup$ Welcome to MSE. You'll get a lot more help, and fewer votes to close, if you show that you have made a real effort to solve the problem yourself. What are your thoughts? What have you tried? How far did you get? Where are you stuck? This question is likely to be closed if you don't add more context. Please respond by editing the question body. Many people browsing questions will vote to close without reading the comments. $\endgroup$ – saulspatz Oct 23 '19 at 14:34
  • $\begingroup$ For part a , I get the expected number is 3. Is it correct? In my thinking that Let X be the number of children. And if the first child is boy or girl, then (X-1)~geom(1/2). Thus, E(X-1)=2. And E(X)=1+E(X-1)=1+2=3. For part b , how to start ? $\endgroup$ – dark chocolate Oct 23 '19 at 14:41
  • $\begingroup$ put it in the question body not the comments $\endgroup$ – Legorhin Oct 23 '19 at 14:43
  • $\begingroup$ Please edit your question to include your reasoning. Also, please use MathJax to format your posts. For a start, enclose your formulas in $ signs. $\endgroup$ – saulspatz Oct 23 '19 at 14:44
  • $\begingroup$ You are right on part a. On part b, how long until the first boy? How long after that until the second? $\endgroup$ – saulspatz Oct 23 '19 at 14:47
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Second Question

Using the hint, if the first child is female, then the expected number of further children is equal to what it was at the outset, because we are no nearer to the requirement of 2 males being met. So if we expect E children, then in the case of the first child being female, we then expect a total of E + 1 children

If the first child is male, then the expected future children is the same as the expectation for the first male, which can be worked out as $M = 1 + \frac{1}{2}M $

gives M = 2 (you've calculated similar with geometric series in question 1)

so for 2 males

$E = 1 + P(first female)E + P(first(male)) \times 2$

$E = 1 + \frac{1}{2}E + \frac{1}{2} \times 2$

$ E = 4 $

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  • $\begingroup$ Thank you , by reading your solution I am understand part b better. $\endgroup$ – dark chocolate Oct 23 '19 at 15:16

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