There is a probability problem on the internet that I really don’t understand. Please explain it to me in a way that makes rational sense to me. I am stealing this text from the webpage linked below.
“Mary has 2 children. She tells you that one is a boy born on a Tuesday. What’s the probability the other child is a girl?
Most people think the answer is 50%. People familiar with the boy or girl paradox may guess 66.6%. But in this particular variation, the answer is about 51.8%. So people are wondering: where are the numbers coming from?”
On this webpage (which also has a video), he goes through the math to show why it is 14/27 or ~51.8%. And just by the way, it assumes the boy/girl birth ratio to be 50%.
I understand the classic answer being 66.6%. I really do not understand how stating that the boy was born on a Tuesday changes the answer. The boy had to be born on a day. There is nothing special about Tuesday. So if we replace Tuesday with any other day, the math will end up the same (51.8%). So if the answer is the same for any day, it will always be the same, which conflicts with the general answer of 66.6%.
Please someone help me make rational sense of the answer!
Firstly in order to reframe the puzzle properly, you should ask her whether she has at least one boy born on a Tuesday, the fact she’s giving the information herself skews the reasoning for why she might say it.
There are 27 equally likely different ways that there can be at least one Boy (Tue)
1/27 the other is a boy (tue)
14/27 it’s a girl 7 times older, 7 times younger
12/27 its a boy (not tue) 6 times younger, 6 times older.
14/27=51.85%
It’s the same logic as the other problem, where removing the chance of both children being girls or (boys not born on Tuesday) increases the odds of the subsection “girls”
I feel like the responses are just restated the probability math, which I sort of follow.
It still doesn’t make sense to me. Can someone respond to my “any day” proposal.
Instead of her saying “he was born on a Tuesday” she could have said Sunday or Monday. If we use those days instead we get:
Boy (born on Sunday): Chance of other child being a girl =51.8%
Boy (born on Monday): Chance of other child being a girl =51.8%
Boy (born on Tuesday): Chance of other child being a girl =51.8%
Boy (born on Wednesday): Chance of other child being a girl =51.8%
…and so on.
So for every day of the week, the chance of the other child being a girl is 51.8%. But that conflicts with the general answer of 66.6%. Can someone help me reconcile these answers?
The standard resolution to this “paradox” is that the probability depends on the exact assumptions of the underlying probability space and selection procedure. So, for example, having a special phonebook which lists all the 2-child families where one of the kids is a boy born on a Tuesday might, not too surprisingly, not lead to the same calculation as randomly knocking on some door and asking the kid who comes out on what day he was born.
There is, in fact, a section on Wikipedia about the “boy or girl paradox” illustrating why, under those assumptions, the more (at first glance seemingly irrelevant) information you give the lower the probability that the other child is a girl:
ETA: largely ninja’ed by @DPRK’s second post that I didn’t see until I finished this one.
Nope. It looks like she could have said that. But she could not have done that and also tell the truth. Because to do so, she’d have needed to have 7 boys, 1 per day, and also simultaneously have just one boy.
Once you drop down to just [specific day of week regardless of which specific day of week], now you’re in 51.8 space, not 66.6 space.
The answer is different again if she had said “I have one boy born on Tue or Thu but I’m not telling you which.”
@DPRK really nailed it conceptually. Unsurprising given his expertise.
Both the puzzle setter and the puzzle solver need to be very precise about what is known, what is unknown, and what the possibility space really is. Almost any reductive thinking by the solver amounts to removing essential information and thereby coming to a wrong conclusion. Of course for puzzle-making, that amounts to inviting people to bite off on the easy but wrong answer. So of course lots of puzzles exploit that aspect.
That’s also why so many internet puzzles are screwed up messes. The setter doesn’t know what they’re doing, or it gets copied with details unwittingly changed. Or details deliberately changed to hide the plagiarism, but the changer doesn’t recognize their changes to the problem trigger changes to the answer too.
Like Socrates, I have to deny any expertise, but thanks
The real point of paradoxes is that they force you to example the underlying assumptions formally: Russell’s Paradox, the Coastline Paradox, the Banach–Tarski Paradox, you name it. If you do not do it, there may not be a well-defined answer. Another famous probability one is, "Randomly break a stick into three pieces. What is the probability that the three pieces form a triangle?
In this problem, as the Wikipedia link explains, under the “traditional assumptions” of this problem, if the extra information about the boy has probability \varepsilon, where, for example, \varepsilon=1 if you just say “Mary has two children, one of whom is a boy” but \varepsilon=1/7 if you say the boy was born on a Tuesday, then an application of Bayes’s Theorem results in a conditional probability of \displaystyle\frac{2-\varepsilon}{4-\varepsilon} that both children are boys.
Again, to backtrack, there is a difference between “Mary has two children, one of whom is a boy” and (under reasonable assumptions, again) seeing Mary with a boy and her telling you that is one of her two children.
There’s also the (possibly inadvertent, possibly deliberate) ambiguity in the first sentence: “Mary has two children, one of whom is a boy”. Which can reasonably be interpreted as either
Mary has two children, exactly one of whom is a boy
Mary has two children, at least one of whom is a boy
The former is the more formal “mathy” reading, while the latter is the more common conversational English reading. Which did the puzzle setter mean? Does that person even know what they did?
The 1/14 chance that the other child is also a boy born on Tuesday is the key to it. It excludes the possibility that both children are girls born on Tuesday and everything else is symmetrical between boys and girls. The same logic as the non-tuesday problem applies, just with a narrower range of exclusions it will be a lot closer to 50-50.
