Baby Boomers

[Ghost Post]

Is there an error in the baby boomers puzzle or am I missing something and am about to be lambasted.

You say the expected number of boys is 1 per family and that for girls is 0.5 (this is the number of expected girls per family and not the proportion as is implied). This would suggest a total population that was not 50:50 male:female, which the answer suggests.

Please could someone explain this problem to me,

Vince

[mathgrant]

It was a miscalculation!!!!!!!!!!!!!! Get over it!!!!!!!

[Ghost Post]

Wanker! I asked a simple question, yet the first answer stated the question was badly worded; which is vague. I therefore asked the same question at the forum. I am new to this site so do not know what has been said in the past about this problem.

Vince

[Chuck]

It appeared that some people didn't understand the problem and some disagreed with the solution. There was a long debate about it. As I see it, the problem states that each child has a 50% chance or being a boy and 50% of being a girl, so the number of boys will end up being approximately equal to the number of girls. It doesn't really matter how you calculate it. If some set of complex calculations disagrees with my solution there must be an error in it, or more likely a misunderstanding of the problem.

Anyone who still disagrees can now start the debate again.

[Ghost Post]

[mathgrant]

Pretend we have infinitely many families. To determine Family 1's children, we flip a coin, adding a girl for each tail and the boy for a head. Suppose we get TTTH. We write that down. Now we do the same for each family, and get something like this:

TTTH/H/H/H/H/TTTTTH/TTH/H/TTTH/H/TH/H/TH/TTTTTH/. . .

If we have infinitely many families, then we can ignore the bars:

TTTHHHHHTTTTTHTTHHTTTHHTHHTHTTTTTH. . .

So now we have an infinite sequence of heads and tails. Clearly half are heads and half are tails. So returning back to the infinite families, we see there are as many boys as girls. This is also true with a finite number of families.

[SeanJuan*]

I reached the right solution, but at first the answer listed, while agreeing with mine, confused me because the math looked wrong. The error is in the equation. They show:

"And so... Therefore the expected number of daughters is (.50 * 0 daughters) + (.25 * 1 daughter) + (.125 * 2 daughters)... = 0.5 daughters. "

But already just from the parts of the sequence shown (BEFORE the ellipsis) we have 0.5. What it SHOULD have said is "=1.0 daughters" (or approaches). Since every child-bearing scenario also has 1 son, this gives the desired 50/50 answer.

[SeanJuan]

There, now I am registered so I can track this thread.

[Jack_Ian]

No time to contribute to the discussion at the moment, other than to supply a link to the original puzzle: Baby Boomers and the Given Solution.

[Zag]

[SeanJuan]

Well, without really sinking a great deal of thought into this (so please, correct me if I'm wrong and I probably won't put up too much of a fight), but I don't think the multiple births (I'm assuming you mean twins, triplets, etc.) would have an impact.

The problem states that once a son is born, the parents will stop reproducing. Now i'm not going to explore the angle that one twin (or triplet, etc.) would have to come out first. Let's just say they all are born simultaneously for point of argument. This is not to say that they shouldn't be looked at as "first" twin and "second" twin, it's just to get around the wording of the question.

So let's consider the twins situation first. If a couple has twins, there are only four scenarios (that we're going to consider, defects aside).

First: Girl, Second: Girl
First: Girl, Second: Boy
First: Boy, Second: Girl
First: Boy, Second: Boy

Now, in the first scenario of two girls, it is irrelevant that twins were had, the couple will keep having children. A similar argument may be made for the second scenario; it doesn't matter that they're twins as the last child out was the only boy, and they shall not have more kids. So ultimately, only the last two scenarios matter, where

a boy has been born but more children are still to come out. As seen here, that is a boy half the time, and a girl half the time, consistent with the answer.

Again, apply this logic to triplets, quadruplets, etc. The only data that matters is when a boy is had in any position other than the last one, and from that point on in the birthing process, it's 50/50 as to whether the next child is a boy or a girl still.

At least, that's how I see it.

[Zag]

Excellent point! I am convinced. My assumption that it doesn't increase the expectation for girls was incorrect, since it allows a family to have a girl after having a boy (as you pointed out).

By the way, welcome to Grey Labyrinth! Please come join us in Visitor Submitted Puzzles and Visitor Submitted Games, too!

[SeanJuan]

Thank you. Just found this site a couple of nights ago and have been enjoying the mental exercise.

[jesternl]

enjoy your stay then

[ralphmerridew]

It allows families to have a girl after a boy, but it also allows families to have multiple boys.

[SeanJuan]

I suppose a much simpler way to rule out the multiple birth scenario is as follows: Without any sort of restriction, the odds are 50/50 boy/girl. The multiple birth scenario would in some manner bring the restricted number closer to that 50/50, and the number would fall between.

But, whether intuitive or not, the answer to the original problem was still that 50/50, so whether it carries weight or not, the numbers already are the same and won't need to meet in the middle.

[Zag]

Right. By exactly the same amount, as SeanJean pointed out. I think you needed to read further back.