## Wednesday, February 13, 2019

### A Country Seeking Males (puzzle)

Today re-running another favorite puzzle, from 6 years ago.  I took it from Richard Wiseman who stated it this way:

"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 it's 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?"

What I like about this puzzle is that (in my experience) it tends to split people into two groups: those who see the solution fairly quickly and think it quite obvious, and those who can barely believe the solution initially when they hear it, and require convincing!

Wiseman’s original post (with its 279 comments) is here:

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The answer is 50%.  One of the simplest explanations from Wiseman’s 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 (allowing for minor variation when you have an odd no. of families).
Another way to look at it is that in the initial step (above) you end up with families having an overbalance of 64 boys; all the remaining steps simply yield enough girls to counter that initial imbalance.
The wording is what makes it tricky for some, who falsely imagine it implying that while no family ever has more than one son, potentially a family could have, say, 1 million daughters before having a son (not so, unless you started with mathematically enough families to allow  for such; in which case you'd still end up with 50/50).