This market is just a math riddle I found intersting
Your know your friend has 2 children, one of which is a boy born on Tuesday.
What is the probability that his second child is also a boy ?
We suppose equal probability of birth for girl and boy and equal probability of being born in any day of the week.
This market will resolve in one month.
I will not participate.
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@cloontia from the information given we don't necessarily know the boy was born first or second, just that the son was born on a non specific Tuesday. Even taking into account that the the son was born on the Tuesday the question was posted, there is no information saying weather the Son was the first or second of a set of twins born on that Tuesday.
Therefore the answer has to be 1/3 because the only combination we can eliminate is the the possibility that the 1st and 2nd child are both girls.
@cloontia Let's follow the logic of this answer in an extreme case. 100 children, 99 known boys. There are 2^100 permutations. We can eliminate any answer with more than one girl. There's 100 children that the girl could be, plus one instance where they're all boys. That's 101 possibilities, 100 of which the final child is not a boy. A less than 1% chance the final child is a boy. So we can confidently say that by this logic, the more known sons there are, the less likely an additional child is to be a boy.
Something seems wrong with this, though. Because don't you only have a 1/100 chance of knowing that there's 99 boys if there was a girl? You're infinitely more likely to know there's 99 boys if it turns out that they were ALL boys.
That, and it REEKS of the gambler's fallacy. "There's 99 boys, which is super improbable, so surely the last one MUST be a girl, right?"
@abledbody Thank you for illustrating my point, perhaps inadvertently! Each event is absolutely its own event like a coin flip.
The wording of the question is why the ANSWER: 13/27 is INCORRECT:
The first child is born on Tuesday, that could be implying that it was born on the Tuesday of the week the question was asked. Therefore the second child would have to be a twin most likely also born on Tuesday. It seems unlikely in any real world scenario that someone would declare " I had a son born on Tuesday and not mention the twin also born on Tuesday or potentially a minute after midnight on Wednesday". With this level of ambiguity really all we can definetely say is true is this:
-there are 2 kids
-one is a boy
-the boy was born on an unknown Tuesday
-no birth order can be established
With those parameters the answer should be 13/41 (or 1/3). Since 13/41 wasnot an option, 1/3 is the correct answer of those presented.
What @btwsbed most likely meant to say was "what is the probability that the first child is also a boy if the parent just had a boy last Tuesday"
Unfortunetly that is not the question that was asked
here is a proof:
@cloontia But can you prove "Girl and Boy" and "Boy and Girl" are actually separate permutations, each with the same probability as "Boy and Boy"? This proof is contingent on that assumption.
@abledbody Thank you for illustrating that point @abledbody, thats exactly right, and exactly my point. So the only permutation we can count out is Girl and Girl
What is clear is the wrong answer was the one given credit for and that @btwsbed had meant to ask the question:
"what is the probability that the first child is also a boy if the parent just had a boy last Tuesday"
Instead of giving credit for the actual question asked:
"What is the probability that his second child is also a boy ?"
There seems to be no oversight or accountability if someone meant to ask one question but accidentally asked another and is unwilling to alter the results for the the actual answer to that question.
A fun way to think about it is by using the probability monad, I give an example here: https://gist.github.com/dignissimus/84dadc6a4ee54128f43c2b49a79ac7c0
"second child" or "other child"? "is also a boy", or "is a boy"? Are we talking the second child to be born, or the child which isn't the one we know is a boy? Also, who cares if they were born on Tuesday? The order in which they're born is completely random. This question is so poorly phrased. There's no way to predict how the author will resolve, but in either case, anyone who bet yes on 13/27 has either heard the complete question somewhere else, or is confidently wrong about probability.
The answer actually depends on how we find out that one of the children is a boy and is born on Tuesday.
If we ask the dad "is at least one of your children a boy and born on a Tuesday yes or no?" and he says yes then it is 13/27.
If we see him with a boy and ask the boy "which day of the week were you born?" and he says Tuesday, then it is 1/2.
@btwsbed yeah I mean, both of those are compatible with the way the question was framed, so I guess the answer should be "not enough information" or something.
-Snip-
How is this even meant to resolve? A mathematical proof? A monte carlo experiment? Census bureau data? Within what error margin would one even interpret the census data? How many real world variables are we meant to take into account for our probability estimation?
Assuming we keep this problem in math land, this is a well-known problem with a well-known solution, except does this market exist purely to educate people on the quirky answer or are you going to claim it's 1/2?
My apologies if I wasn't clear on this market (I'm new to Manifold), it's just a math problem I came across recently and thought it would be interesting to see if anyone would assert an answer other than the correct one and why.
I put 1/2 just to have at least one answer and not to keep it with others (idk if it was a good idea).
@AngolaMaldives I hope that is “other” and not in birth order. As the question is if his second child is “also” a boy. Scary mkt to trade on…