Solution to warm-up:
Let us represent boys by 1's and girls by 0's. There are eight equally likely scenarios for three children:

000
001
010
011
100
101
110
111

In only one case will there be no boys (000). That will happen to only one family in eight. So, 7/8 of the families will have a boy.

One might wonder whether it is legitimate to count as different families with the same set of real children but different sets of phantom children. For example, 100, 101, 110 and 111 all end up as a family with one boy. The best way to think about this is to imagine that the phantom children are in fact real but are sent abroad to a less sexist culture. In that case, every family has three children, but the family possibilities that remain in Machudo are just boy; girl, boy; girl, girl, boy; and girl, girl, girl.

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7 Comments

1. 1. dhbarnett 12:35 AM 8/1/08

   SPOLER:
   
   Is this right? According to the instructions families stop having children when they have a boy, but you have 010 and 011 as a posibility. Would the set not be
   000
   01
   1
   

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2. 2. PilotPrecise 03:21 AM 8/1/08

   I figured that differently. Complete families would be 000, 001, 01, and 1. My answer was: 3/4

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3. 3. PilotPrecise in reply to PilotPrecise 03:33 AM 8/1/08

   And now I see why I was wrong. :)

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4. 4. PilotPrecise in reply to PilotPrecise 03:37 AM 8/1/08

   Of course, I mistook the nature of the problem I was figuring and came up with the wrong answer. :)

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5. 5. spork 11:35 AM 8/1/08

   The more interesting question would be this:
   
   Suppose every family is forced to have children until they have a son, and then are sterilized. What would be the average number of children per family? (What makes this interesting is that there is a non-zero chance that a family will have thousands of daughters before having a son, and this must be accounted for in the calculation.

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6. 6. felipecarvajal 04:29 PM 8/4/08

   The correct answer as defined by the wording in the problem is .75. Since the phrase "so they would stop having children after they had their first boy" would eliminate the possibility of brothers. Only the following families would result:
   "1" a boy on the first try
   "10" a boy on the second try
   "100" a boy on the third try
   "000" no boys, but a complete family
   
   The phrase "so they would stop having children after they had their first boy" would have to be omitted in order to allow the preceding statement , "Their eldest son had a chance of becoming king" to make any sense. For 7/8 to be the correct answer, we must eliminate the following two phrases. "so they would stop having children after they had their first boy", and "or had a boy", since either would alter the three child parameter.
   
   ><)))> Felipe

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7. 7. minasto in reply to dhbarnett 06:44 AM 8/8/08

   What about if every two families from every Eight families Don't have any boy? ........ this is in control of God..........

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