Puzzle Corner

As Laird's IQ puzzle was such a success, perhaps we should have a dedicated puzzle thread.

Here's one from the Skeptical Intelligencer: "Do boys have more sisters than girls do?" (For the purposes of the puzzle, assume each child is equally likely to be a boy or a girl, independently of the sex of its siblings.)

if I understand the question right then yes, boys should have more sisters than girls.

If everything ie equal the question can be written where there's 2 boys and 2 girls, all siblings. Each boy has 2 sisters, but each girl only has 1.

if I understand the question right then yes, boys should have more sisters than girls.

If everything ie equal the question can be written where there's 2 boys and 2 girls, all siblings. Each boy has 2 sisters, but each girl only has 1.

I think it actually works out even. 

Show contentSpoiler:

Yes there is an imbalance when it's 2 boys and 2 girls. But when it's all girls, it adds up to 12 sisters (each of the 4 girls has 3 sisters). Taking all possible combinations into account adds up to the same number of sisters.

Linda

[Laird]

I think Linda's idea to enclose solutions or commentary on them in a spoiler is a good one, so here are my comments/analysis on her own analysis in a spoiler:

Show contentSpoiler:

I think it actually works out even.

Yes. This seems to be the case.

Yes there is an imbalance when it's 2 boys and 2 girls. But when it's all girls, it adds up to 12 sisters (each of the 4 girls has 3 sisters).

Right, and this latter is counter-balanced also by cases such as 3 boys and 1 girl, in which that girl has no sisters but each of the three boys has one, for a total of three.

Taking all possible combinations into account adds up to the same number of sisters.

Yes, that seems to be correct. Here are the tables for 2, 3, and 4 siblings:

Code:

    G sisters    B sisters
GG    1,1
GB            1
BG            1
BB
Avg:    1        1

    G sisters    B sisters
GGG    2,2,2
GGB    1,1        2
GBG    1,1        2
GBB            1,1
BGG    1,1        2
BGB            1,1
BBG            1,1
BBB    
Avg:    12/9 = 1 3/11    12/9 = 1 1/3


        G sisters    B sisters
GGGG    3,3,3,3
GGGB    2,2,2    3
GGBG    2,2,2    3
GGBB    1,1        2,2
GBGG    2,2,2    3
GBGB    1,1        2,2
GBBG    1,1        2,2
GBBB            1,1,1
BGGG    2,2,2    3
BGGB    1,1        2,2
BGBG    1,1        2,2
BGBB            1,1,1
BBGG    1,1        2,2
BBGB            1,1,1
BBBG            1,1,1
BBBB    
Avg:    48/28 = 1 5/7    48/28 = 1 5/7

By both totals and averages, girls have the same number of sisters as boys.

[Laird]

Apologies. My tables are incorrect. Let me adjust. You seem to be right after all, Linda.

[Laird]

Fixed. For a bit there I was thinking this was something like expectation bias! But it's not.

I think it actually works out even. 

Show contentSpoiler:

Yes there is an imbalance when it's 2 boys and 2 girls. But when it's all girls, it adds up to 12 sisters (each of the 4 girls has 3 sisters). Taking all possible combinations into account adds up to the same number of sisters.

Linda

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?

[Laird]

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."

That doesn't mean an equal split though. It's a likelihood, so you could get different splits at given likelihoods. e.g. see my tables above (the last one in particular, with four siblings, where "B" represents "boy" and "G" represents "girl": the likelihood of an equal split is 6/16 = 3/8, because of all sixteen equally likely combinations, only four rows contain an equal split of boys and girls (rows 4, 6, 7, 10, 11, 13).

So, 3 times in 8 you will get an equal split - not 100% of the time as you assumed.

That doesn't mean an equal split though. It's a likelihood, so you could different splits at given likelihoods. e.g. see my tables above (the last one in particular, with four siblings, where "B" represents "boy" and "G" represents "girl": the likelihood of an equal split is 6/16 = 3/8, because of all sixteen equally likely combinations, only four rows contain an equal split of boys and girls (rows 4, 6, 7, 10, 11, 13).

So, 3 times in 8 you will get an equal split - not 100% of the time as you assumed.

Yeah that's true, but then there's literally no point including that assumption in the question if that's the case since none of it's implications ever come into play. I'm not calculating likelihood of N sibling being male or female or anything. So it's a statement that can only be thrown in to either constrain the possible combination of siblings, which is how I took it, or to be purposefully misleading and pointless. Which would just be bad question design as it makes what you're even asking unclear.

[Laird]

there's literally no point including that assumption in the question if that's the case since none of it's implications ever come into play.

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.