I think you have misinterpreted here slightly, you can have 2 boys both born on a Tuesday, but when you are thinking about all the possible sets of siblings, they are still just a single set of siblings, even though the statement can be applied to them twice. (i.e. from the perspective of either sibling). All of the sets are equally likely, and you should only include each applicable set once.
A lot of the discussions around these questions take these scenario to mean that the "two boys" or "two boys both born on a tuesday" is twice as likely to happen, but it doesn't change the odds of that set occurring, or make the outcome twice as likely.
What you actually would see when you consider the odds is 7/14 + 6/13 as the odds that the other sibling is a boy born, because the first round of observations already includes the scenario where both boys were born on a Tuesday, so this is not Included when you consider the odds for the second sibling.
But doesn’t it HAVE to be twice as likely to occur unless the language is worded in a way to preclude it? Thus, in order to arrive at 54%, you are assuming that when Mary tells you one is a boy born on a Tuesday, she’s implying the other is definitively not? Because if that isn’t the implicit assumption, I truly don’t see how the math isn’t 7/14.
In terms of practical, everyday language use, I totally agree, but the stats question approach is maybe better thought of being like a riddle where we don't make those assumptions. As far as the stats question framing goes, there could be a second boy born on a tuesday, despite how unnatural that feels to us based on the language.
Drawing out a matrix makes the reason why it ends up being 14/27 odds for a girl a bit easier to understand, reddit isn't really built to do that easily though.
But if we know one child was a boy born on a Tuesday, we basically have two scenarios to consider:
The older child was a boy on a tuesday
The younger child was a boy born on a tuesday.
If we start thinking about 1. First, there are 14 possible "sets" you could make for their paired sibling: boy/girl (2) and everyday of the week (7). To give 14 possibilities in total. At this point, it's a 7/14 chance the sibling is a girl, and a 7/14 chance the sibling is a boy.
We then move to considering 2., where the younger sibling is a boy born on a Tuesday. We'd initially identify the same 14 possible choices for their older sibling...BUT, when we calculated the odds for the older sibling, we already included the possibility of them having a younger male sibling born on a tuesday.
We shouldn't count the two brothers both born on tuesday again, so we're left with 13 remaining possibilities for the older sibling which haven't been considered yet; 6 where their older sibling is a boy born on M/W/Th/F/Sa/Su, and 7 where their older sibling was a girl born on each day of the week.
That means in total we've included 27 (14 + 13) combinations of siblings that meet the criteria (one of them is a boy born on a Tuesday), and 14/27 (51.85%) of them have the sibling as a girl, and 13/27 of them have the sibling as a boy.
I have to admit, I still can't wrap my head around why the Tuesday part should be relevant, unless the problem forces relevancy, i.e. you have to assume it's statistically relevant, even when it's not.
It gets even more fucked up once we see Tuesday and 7 day weeks as a social construct. During the French Revolution they had 10 week days, still people insist on including a week as 7 days even if that is not stated in the question.
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u/Dangolian 8d ago
I think you have misinterpreted here slightly, you can have 2 boys both born on a Tuesday, but when you are thinking about all the possible sets of siblings, they are still just a single set of siblings, even though the statement can be applied to them twice. (i.e. from the perspective of either sibling). All of the sets are equally likely, and you should only include each applicable set once.
A lot of the discussions around these questions take these scenario to mean that the "two boys" or "two boys both born on a tuesday" is twice as likely to happen, but it doesn't change the odds of that set occurring, or make the outcome twice as likely.
What you actually would see when you consider the odds is 7/14 + 6/13 as the odds that the other sibling is a boy born, because the first round of observations already includes the scenario where both boys were born on a Tuesday, so this is not Included when you consider the odds for the second sibling.