Puzzle Corner

But they do come into play: in calculating the probability of an equal split of 3 in 8, I assumed that each row of the table was equally likely to occur, which in turn was based on that assumption. If the different rows occurred with different likelihoods (i.e., if the likelihood of a child being a boy or girl was not equal), then the probability would not have been 3 in 8 - it would have been some other value.

Then we answered different questions, it didn't sound like actual statistics were asked for so I didn't use any. it sounded like a very straight forward single 50/50 split question where I could set any number of total kids I wanted and I only had to assume they were split evenly because if I had 20 kids, 19 of which were boys, it's going to really look like I'm not assuming equal distribution. But I guess that just means I failed the test.

[Laird]

But they do come into play: in calculating the probability of an equal split of 3 in 8, I assumed that each row of the table was equally likely to occur, which in turn was based on that assumption. If the different rows occurred with different likelihoods (i.e., if the likelihood of a child being a boy or girl was not equal), then the probability would not have been 3 in 8 - it would have been some other value.

To elaborate, because calculating that probability is not strictly part of the puzzle: if the likelihoods of each child being a boy or a girl were not equal, then I could not simply sum up the numbers in the two data columns of my table: I would need to work out probabilities first to multiply them by.

[Laird]

it sounded like a very straight forward single 50/50 split question where I could set any number of total kids I wanted and I only had to assume they were split evenly because if I had 20 kids, 19 of which were boys, it's going to really look like I'm not assuming equal distribution.

All you have to assume is that all permutations of 20 kids are equally likely. So, you could have 19 boys of 20, but you could also have 19 girls of 20.

All you have to assume is that all permutations of 20 kids are equally likely. So, you could have 19 boys of 20, but you could also have 19 girls of 20.

Yeah but I'll never be able to definitively answer if boys have more sisters than girls, because statistics can only say what's likely to happen rather than what's actually happening in any single instance, so it's an answer that will always be in superposition. I was asked for a binary yes or no, so that's how I did the question.

[Laird]

You can definitively answer the question by considering all possibilities (as equally likely) and then taking averages, which is the approach suggested by Linda and which proved her to be correct.

You can definitively answer the question by considering all possibilities (as equally likely) and then taking averages, which is the approach suggested by Linda and which proved her to be correct.

I'm not debating my incorrectness, I already know I failed. I just think the question was worded badly. But whatever, that's also probably part of it.

[Laird]

I'm not debating my incorrectness, I already know I failed. I just think the question was worded badly. But whatever, that's also probably part of it.

No worries.

I did no counter balancing calculations because the parameter of the question was to assume equal split,

"assume each child is equally likely to be a boy or a girl, independently of the sex of its siblings."

so that's what I did. it didn't sound like it was asking for all possible combinations, because if it was, why not just say that?

"Equally likely" is not the same as telling you to consider only combinations where the numbers of boys and girls were equal. It's telling you how many times you will see each possible combination (including combinations where the numbers are equal and those where the numbers are not equal) based on probability. This becomes obvious if you apply the question to a family with 3 children.

If the question is not asking you to take all possible combinations into consideration, then it's not a puzzle. It's just a very easy counting question.

Linda

Show contentSpoiler:

According to the Skeptical Intelligencer, the right answer is that boys and girls have the same number of sisters. Though I agree that the question was badly worded, because of course it should have said "on average." (In my own defence I just quoted the part in "..." straight from the Intelligencer puzzle column.)

As Laird implied, the lurking fear is that there is some subtle averaging artefact going on, with families with more boys (and therefore fewer sisters) contributing more to the average for boys and less to the average for girls. But the direct calculation confirms that's not the case. Alternatively, we can start by just thinking about the eldest child, and it's clear that on average half the younger children will be boys and half girls, regardless of whether the eldest child is a boy or a girl. Then we can do the same for the second child and so on.

The caveat about equal probabilities independent of the sex of the other children is necessary. For example, if boys and girls were equally likely, but younger children always had the same sex as the first child, then boys would never have any sisters at all.

I wonder whether I should also have stipulated that parents mustn't be allowed to stop having children based on the sex of the children they've had so far?

Show contentSpoiler:

According to the Skeptical Intelligencer, the right answer is that boys and girls have the same number of sisters. Though I agree that the question was badly worded, because of course it should have said "on average." (In my own defence I just quoted the part in "..." straight from the Intelligencer puzzle column.)

As Laird implied, the lurking fear is that there is some subtle averaging artefact going on, with families with more boys (and therefore fewer sisters) contributing more to the average for boys and less to the average for girls. But the direct calculation confirms that's not the case. Alternatively, we can start by just thinking about the eldest child, and it's clear that on average half the younger children will be boys and half girls, regardless of whether the eldest child is a boy or a girl. Then we can do the same for the second child and so on.

The caveat about equal probabilities independent of the sex of the other children is necessary. For example, if boys and girls were equally likely, but younger children always had the same sex as the first child, then boys would never have any sisters at all.

I wonder whether I should also have stipulated that parents mustn't be allowed to stop having children based on the sex of the children they've had so far?

Indeed, this doesn't even take into account the children's preferred genders!