The question is asking for the fraction of girls in the whole country, which would be the total number of girls divided by the total number of children, across all families.
Of course that’s a random number, so for it for to be a meaningful question at all you have to average over many trials or similar countries.
@yebellz helpfully modeled children as coin flips and simplified to the case of a country with a single family:
And I suggested simplifying further to limit it to a maximum of three flips. You have correctly listed the probabilities is the different outcomes.
But now to get the expected fraction of tails we need to average that fraction, not numbers of heads or tails.
I see a trial as a sequence of coin flips ending in heads (or in case we cut off the trial at a certain step, a sequence of that length with only tails). This is the same as a family getting children, up to a certain number of daughters, ending with a son.
This is what I’ve computed in about 5 different ways now: the total fraction of girls in the country = the total number of heads being thrown divided by the total number of coin flips = the expected percentage of daughters in one family = the expected percentage of heads in a trial = 50% of the total population.
None of this is relevant to the question at stake: what is the estimated ratio between daughters and sons, or between heads and tails. We’re not doing statistics here, we don’t have samples. We’re doing probability: considering multiple countries / sessions of coin flips don’t make a difference.
This is what I did: I took the weighted average of the fractions. It’s the same as the ratio between the number of heads and the number of tails, that’s the whole meaning of “fraction”.
Perhaps it would be helpful if you would explain what you expect the answer to be? Since I don’t have a clue what you’re trying to point out.
It’s a crucial difference, as this single-family example shows!
As long as we’re sticking with this example, each row in your table is a single outcome of a trial, a possible country with just one family. The weights are the probabilities of the different outcomes. So you should get 1/2*0 + 1/4*1/2 + 1/8*2/3 + 1/8*1 = 33%.
Since half of the outcomes were 0% tails, the only way to get the average up to 50% would have been to have 100% tails in the other half.
I think you computed something different: the expected number of tails divided by the expected number of total flips.
Hmmmmm… I think I can see what you’re getting at. I think I interpreted the question wrongly, averaging the fractions seems like a very counterintuitive thing to do in general. I would be lying if I said I understand the why behind it, yet, though… I’ve been trying to put it into words for about 30 minutes now, I’ll have to study this a bit more carefully.
For any single family, we should expect the fraction of girls to total children to be 0 • 1/2 + 1/2 • 1/4 + 2/3 • 1/8 + … = ∑ (n-1 / n) • 2-n, which is apparently roughly 0.30, and exactly 1 - ln(2). This does not contradict the intuition that half of the population is female, since those families that fit the expectation of having few daughters also contribute less to the total population. As soon as we account for population size (and average our fractions weighted by size of the family), we get the 50-50 distribution. It therefore seems that if we have a country with large number of families, that the expected fraction should approach 0.5, although I’m struggling with formalising this idea exactly.
This would mean that the answer does depend on whether we allow for families getting arbitrarily many children. If that is allowed, then it’s not unreasonable to allow countries with arbitrarily large number of families, and it quickly follows that the average expected fraction for an arbitrary country must be larger than the expected fraction for an arbitrary country of any fixed (finite) number of families.
Hence, the answer would actually be 0.5 when we consider countries with arbitrarily large numbers of families, but less than 0.5 as soon as we limit our countries to have at most a certain number of families (or put some kind of distribution on the number of families; in fact, I’m not sure if it’s nonsensical to ask the question without first saying how the population size of the countries is distributed).
If that is the case, then yes. It sounds quite reasonable to me that it would be the case, but I’ve spent the whole day thinking about this problem, I need to focus on some other things for a while
You’re buying a baseball and a bat. It turns out the bat costs 1 dollar more than the baseball. In total you pay $1.10. What is the price of the baseball?
$0.05
$0.10
$0.15
$0.20
I don’t know
0voters
I have a red and a blue deck of cards, and taken out some aces and kings from both decks:
Unless those are the only four cards that exist (in the universe that your hypothetical is considering), I cannot not confirm that you are definitely telling the truth, but only either falsify or fail to falsify your claim.
