## Thursday, February 28, 2013

### A Puzzle Re-run

One of the advantages of math blogging is that so much of mathematics never gets old! If you're doing political blogging than what you wrote about a month ago is likely already aged, boring news. But in math so much material is timeless (people still do posts on the Pythagorean theorem).
So being in a puzzle mood today, I've reached back to a post from over a year ago for another of my favorites (and of course the puzzle is older than that). I'll quote largely from that post with a few changes (and apologies to all who are familiar and bored with this classic, but for newbies who've never seen it....):

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Verbatim from Richard Wiseman's blog:
"Imagine there is a country with a lot of people. These people do not die, the people consists of monogamous families only, and there is no limit to the maximum amount of children each family can have. With every birth there is a 50% chance its a boy and a 50% chance it is a girl.  Every family wants to have one son: they get children until they give birth to a son, then they stop having children. This means that every family eventually has one father, one mother, one son and a variable number of daughters.  What percent of the children in that country are male?"
Wiseman's Friday puzzles are frequently devious… but, often once the answer is given and explained, one feels impelled to slap one's forehead and exclaim "DOH!, well, of course!" So perhaps his best offerings are those that, even once explained, are still not totally clear, and generate much ongoing discussion...

This is one such effort, which often generates dissenters when presented, once again demonstrating how tricky/misleading, probabilities can be. I confess to originally requiring extra time to convince myself that 50% was the correct answer. It's one of those quirky puzzles that is patently obvious to many, yet thorny for others (one of the keys, I think, is to remain tightly focused on strict statistical probability, and not let your brain get distracted by what could theoretically happen). Read all about it for yourself at Richard's original post:

(...and peruse as many of the 270+ comments as you care to!)

One of the simplest explanations from his comments (for anyone having trouble seeing it) is just to imagine the statistics for a sample that begins with 128 families (assuming strict 50% chance of a boy or girl at each point):

128 starting families produce 64 boys and 64 girls
next round, the 64 families with girls now produce 32 boys and 32 girls
next round, the 32 families with girls produce 16 boys and 16 girls
16 families with girls produce 8 boys and 8 girls
8 families with girls produce 4 boys and 4 girls
4 families with girls produce 2 boys and 2 girls
2 families with girls produce 1 boy and 1 girl

Total at conclusion: 50% boys, 50% girls

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