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Use Bayes rule to mechanically solve probability riddles

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Re: Use Bayes rule to mechanically solve probability riddles

#21

Post by kgwgk »

FabHK wrote: post #20
kgwgk wrote: post #18

Earlier quoted context omitted.

> many formulations one finds (including the one in her column) As far as I’ve been able to find the text below would be the original question (“the host, who knows what’s behind the doors, opens”) and answer (“the host, who knows what’s behind the doors and will always avoid the one with the prize, opens”). Are you referring to that? https://web.archive.org/web/20130121183432/http://marilynvos... Suppose you’re on a…

Yes, indeed, thanks for digging it up. You see that Marilyn's answer contains the clarified correct specification ("the host, who knows what’s behind the doors and will always avoid the one with the prize..."), but the question does not fully specify it. Thus, ignoring the clarification in the answer (which furthermore pertains to a modified problem), one could interpret the question differently: the host randomly op…

Did any of those readers of her column, including PhD and mathematicians, that declared her solution wrong do so because they objected to the clarification? (Does it make sense to declare the answer wrong ignoring the answer?)

Re: Use Bayes rule to mechanically solve probability riddles

#22

Post by FabHK »

knappa wrote: post #19
sparsely wrote: post #7

For the last one, why does the "born on a Tuesday" information change the result? I don't see how it isn't equivalent to "born on a day", since the day of the week has no connection to the rest of the scenario. I understand why "at least one boy" does matter.

You could definitely replace "Tuesday" with something like that and part of the pedagogical purpose of the problem is for people to question this. The actual effect comes from not distinguishing the boys. That increases the likelihood that at least one of them will be born on any particular day, upweighing the likelihood that there are larger numbers of boys. i.e. You just get, on average, better coverage of boys-bor…

Well, "born on a day" would not convey any information unless it means "during daytime". If that has probability 1/2, the answer would be 3/7. With Tuesday (or, indeed, any other weekday, with probability 1/7), it is 13/27.

Re: Use Bayes rule to mechanically solve probability riddles

#23

Post by AnotherGoodName »

kgwgk wrote: post #2

> You're told that at least one of them is a girl. > Likelihood of at least one girl What the “mechanism” requires is “likelihood of being told that at least one of them is a girl”. Use Bayes rule to correctly solve probability riddles: https://news.ycombinator.com/item?id=45056790 p(both are girls | you're told at least one is a girl) = p(you're told at least one is a girl | both are girls) * p(both are girls) / ( p…

Pretty much all of them are like this and it's actually a terrible article. For those thinking about how can the likelihood of them being a girl can change consider this;

>A family has two children. You're told that at least one of them is a girl. What's the probability both are girls?

"The question writer took all sets of two child families and ruled out the bb case. Then they asked the exact question above" This is 1/3 chance - select gg from [gg,bg,gb]'

vs

"The question writer came across a girl from a two child family, then they asked the exact question above". This is 1/2 chance - select gg from [gg, gg, bg, gb] with gg listed twice since there's two ways to select a girl from that set; ie. coming across a girl is twice as likely to occur from the gg case than it is either gb or bg.

I think that's the clearest wording to get the message across. Either way it's the exact same question but it reasonably has a completely different answer. There's no way to resolve this ambiguity with the question as written. Pretty much all questions here are like this.

Amusingly the Monty Hall problem in it's original form was written to avoid this ambiguity. There's a comment thread above on this. But you know what the author of the article linked here did? They reworded it and added ambiguity similar to the above. The Monty Hall problem needs to specifically state "the host opens a door that he knows does not contain the prize" or it's unanswerable. The author of the article removed this statement without awareness of its importance!

This sounds mean but honestly I really really think this article was written by someone with no education or knowledge on statistics. As in they broke perfectly reasonable questions and added ambiguity to the point they are not answerable as written in an article trying to demonstrate how easy stats is.

Re: Use Bayes rule to mechanically solve probability riddles

#25

Post by FabHK »

kgwgk wrote: post #21
FabHK wrote: post #20

Earlier quoted context omitted.

Yes, indeed, thanks for digging it up. You see that Marilyn's answer contains the clarified correct specification ("the host, who knows what’s behind the doors and will always avoid the one with the prize..."), but the question does not fully specify it. Thus, ignoring the clarification in the answer (which furthermore pertains to a modified problem), one could interpret the question differently: the host randomly op…

Did any of those readers of her column, including PhD and mathematicians, that declared her solution wrong do so because they objected to the clarification? (Does it make sense to declare the answer wrong ignoring the answer?)

I think they didn't notice the significance of that clarification (which, again, was given in the context of a modified problem). Reading the exchanges again, though, I see that many respondents were not only mistaken, but also impolite or sexist in accusing Marylin of being wrong.

If you specify the problem carefully, anyone with some training in probability should get the answer right. But clearly back then it was a head scratcher.

I wonder whether there've been other problems like that, or we will encounter similar ones: elementary and easy to understand, yet many people get it confidently wrong, until the correct solution permeates through culture.

Re: Use Bayes rule to mechanically solve probability riddles

#26

Post by fenomas »

FabHK wrote: post #20
kgwgk wrote: post #18

Earlier quoted context omitted.

> many formulations one finds (including the one in her column) As far as I’ve been able to find the text below would be the original question (“the host, who knows what’s behind the doors, opens”) and answer (“the host, who knows what’s behind the doors and will always avoid the one with the prize, opens”). Are you referring to that? https://web.archive.org/web/20130121183432/http://marilynvos... Suppose you’re on a…

Yes, indeed, thanks for digging it up. You see that Marilyn's answer contains the clarified correct specification ("the host, who knows what’s behind the doors and will always avoid the one with the prize..."), but the question does not fully specify it. Thus, ignoring the clarification in the answer (which furthermore pertains to a modified problem), one could interpret the question differently: the host randomly op…

I don't think this complaint stands at all. The original question is stated colloquially, not formally, and vos Savant's interpretation of it is the one that's consistent with the question's language and with how game shows work. Even if other interpretations are possible, hers is the one a normal reader would make, and the one the letter-writer apparently intends.

> And that ambiguity has given rise to so much spilled ink (and, by the way, the misconception that statistic professors don't understand probability theory).

I don't think this stands either - the letters quoted don't say anything about the distinction you're making. Unless they're all fictional or selectively edited, a bunch of PhDs really did get the puzzle wrong.

Re: Use Bayes rule to mechanically solve probability riddles

#27

Post by kgwgk »

FabHK wrote: post #25
kgwgk wrote: post #21

Earlier quoted context omitted.

Did any of those readers of her column, including PhD and mathematicians, that declared her solution wrong do so because they objected to the clarification? (Does it make sense to declare the answer wrong ignoring the answer?)

I think they didn't notice the significance of that clarification (which, again, was given in the context of a modified problem). Reading the exchanges again, though, I see that many respondents were not only mistaken, but also impolite or sexist in accusing Marylin of being wrong. If you specify the problem carefully, anyone with some training in probability should get the answer right. But clearly back then it was…

> [that clarification] was given in the context of a modified problem

I don’t understand what do you mean by that. The clarification was part of the original answer in the context of the original question. The clarification was stated again and again and again. If that’s a “modified problem” so be it but that was the problem being discussed with her readers.

Re: Use Bayes rule to mechanically solve probability riddles

#29

Post by michaelcampbell »

kgwgk wrote: post #18
FabHK wrote: post #15

Earlier quoted context omitted.

To be fair, I think this problem needs to be formulated very carefully and specifically [0], because the "correct" answer is predicated on that, and many formulations one finds (including the one in her column) are not that. [0] It has to be specified that a) Monty knows what's behind which door, and b) he will on purpose always open a door such that there's a goat behind it.

> many formulations one finds (including the one in her column) As far as I’ve been able to find the text below would be the original question (“the host, who knows what’s behind the doors, opens”) and answer (“the host, who knows what’s behind the doors and will always avoid the one with the prize, opens”). Are you referring to that? https://web.archive.org/web/20130121183432/http://marilynvos... Suppose you’re on a…

> Suppose there are a million doors,...

This is a common way to explain it, but I always found it less intuitive; not sure why.

My go to thought experiment along these lines is rather this:

Adjust the game to where both you and your friend are playing at the same time. You _NEVER_ switch doors, and your friend _ALWAYS_ does, when given the choice by Monty.

Since you never switch, no matter what Monty does you know you're going to win 1/3 of the time. Since you both know Monty is always going to show a "goat" door, ONE of you must win, so your friend MUST WIN ALL THE TIMES YOU DO NOT. Since you win 1/3 of the time, the wins "left" is 2/3 of the time.

Re: Use Bayes rule to mechanically solve probability riddles

#30

Post by actionfromafar »

Shorel wrote: post #5

This Monty Hall problem was asked to Marilyn vos Savant, a woman with an extremely high IQ, who solved it correctly, and many readers of her column, including PhD and mathematicians, declared her solution wrong. Then careful analysis proved her correct. https://en.wikipedia.org/wiki/Monty_Hall_problem#Savant_and_...

A very smart person is named "savant" and "her grandmother's name was Savant; her grandfather's, vos Savant".

https://en.wikipedia.org/wiki/Marilyn_vos_Savant

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