5 ms·

Use Bayes rule to mechanically solve probability riddles

76 points by zaik 1y ago

MontyCarloHall 1y ago
Recent related discussion: https://news.ycombinator.com/item?id=45051798 https://news.ycombinator.com/item?id=45051798
kgwgk 1y ago
> 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 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(you're told at least one is a girl | both are girls) * p(both are girls) + p(you're told at least one is a girl | they aren't both girls) * p(they aren't both girls) ) The solution there assumes that p(you're told at least one is a girl | both are girls) = p(you're told at least one is a girl | they aren't both girls).
- adammarples 1y ago
  Assuming 100% likelihood of a truth telling quiz setter, that bfomes down to population level statistics of male vs female?
  - kgwgk 1y ago
    This is not about someone choosing to lie, it’s about someone choosing what true thing to say. If “you're told that at least one of them is a girl” was intended to mean “you ask whether at least one of them is a girl and you’re told that at least one of them is a girl” it was pretty easy to make it clear. In that case it’s straightforward that p(you're told at least one is a girl | both are girls) = p(you're told at least one is a girl | only one is a girl) = 1 (with just the mild and reasonable assumption that you’re not being lied to.) Edit: also the reasonable assumption that your question is unrelated to the number of girls - it’s easy to imagine settings where you asked because of something which may be correlated with the number of girls.
- AnotherGoodName 1y ago
  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.
  - zaik 1y ago
    I did not mention a lot of assumptions. For example my problem statement also does not answer the question if Monty always gives you a choice to switch or only when you picked in a certain way. There are quite a few possible variations, all of which change the answer: https://en.wikipedia.org/wiki/Monty_Hall_problem#Other_host_behaviors https://en.wikipedia.org/wiki/Monty_Hall_problem#Other_host_... In the derivation of the likelihood the assumptions are clearly stated. While my knowledge on statistics is certainly unsatisfactory, your assessment of the education I had is quite wrong. My final exam in order to obtain a Master's degree in mathematics is next week. One of the subjects I chose is Bayesian Statistics, so wish me luck.
rtrgrd 1y ago
Might be hug of death but the load times are horrifically slow.
- fuomag9 1y ago
  For me nextcloud has always been worryingly slow even on my instance
  - FabHK 1y ago
    Is the markdown rendered once on the server and stored as HTML? Then why is it slow? Or is it rendered for each client, or rendered in the client?
- deleted 1y ago
  [deleted]
Shorel 1y ago
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_the_media_furor https://en.wikipedia.org/wiki/Monty_Hall_problem#Savant_and_...
- FabHK 1y ago
  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.
  - kgwgk 1y ago
    > 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://marilynvossavant.com/game-show-problem/ https://web.archive.org/web/20130121183432/http://marilynvos... Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors? [Craig F. Whitaker - Columbia, Maryland] Yes; you should switch. The first door has a 1/3 chance of winning, but the second door has a 2/3 chance. Here’s a good way to visualize what happened. Suppose there are a million doors, and you pick door #1. Then the host, who knows what’s behind the doors and will always avoid the one with the prize, opens them all except door #777,777. You’d switch to that door pretty fast, wouldn’t you?
    - FabHK 1y ago
sparsely 1y ago
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.
- zaik 1y ago
  If you accept the Bayes theorem, the answer is that the likelihood of "At least one boy is born on a Tuesday" is not the same for different numbers of boys. The more boys the more likely the statement is true. Therefore this information is indicative of how many boys Mrs. Chance has.
  - jammaloo 1y ago
    It seems to me that it comes down to how the day of the week was picked. If they picked a random day of the week, and there was only one boy, then there is only a 1/7 chance of a boy being born on that day. If they have one boy, who was born on a Tuesday, and that is why they picked the day, then there is a 100% chance of a boy being born on that day, so no additional information is conferred.
  - deleted 1y ago
    [deleted]
  - fenomas 1y ago
    Puzzles like that one have always seemed dishonest to me. It only makes sense if you start from the conclusion that it's meant to illustrate Bayes rule, and then work backwards to the assumption that the predicate "boy born on a Tuesday" is supposed to be independent of who's being asked about. But in plain English, ".. At least one of them is a boy born on Tuesday" suggests the speaker is giving a fact that was chosen because it's true of the person spoken about - like if the kids were both chosen on Thursday then that would be the day named. And read that way, the Bayes illustration doesn't stand and the "correct" answer makes no sense. To make it honest, it should really be worded like: "Mrs. Chance has two children of different ages. You ask whether at least one of them is a boy born on Tuesday, and you are told yes. What is the probability that both of them are boys?" Or am I missing something?
qwertytyyuu 1y ago
I feel this the case because pretty much all probability riddles are bayes rules problems which is famously not intuitive
0xfaded 1y ago
Just a tidbit for remembering bayes rule: P(A|B)P(B) = P(A,B) = P(B|A)P(A) The familiar forms P(A|B) = P(A,B)/P(B) = P(B|A)P(A)/P(B) immediately follow
- glitchc 1y ago
  Comma is not a defined operator. Do you mean intersection?
  - 0xfaded 1y ago
    It's a joint distribution :) https://en.m.wikipedia.org/wiki/Joint_probability_distribution https://en.m.wikipedia.org/wiki/Joint_probability_distributi...
- bobbylarrybobby 1y ago
  In words, for those who need help with the first line: the chance that A and B both happen is the chance that A happens, and (given that A happened) B happens. (Or vice versa.)

76 points by zaik 1y ago

Recent related discussion: https://news.ycombinator.com/item?id=45051798 https://news.ycombinator.com/item?id=45051798

> 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 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(you're told at least one is a girl | both are girls) * p(both are girls) + p(you're told at least one is a girl | they aren't both girls) * p(they aren't both girls) ) The solution there assumes that p(you're told at least one is a girl | both are girls) = p(you're told at least one is a girl | they aren't both girls).

adammarples 1y ago
Assuming 100% likelihood of a truth telling quiz setter, that bfomes down to population level statistics of male vs female?
- kgwgk 1y ago
  This is not about someone choosing to lie, it’s about someone choosing what true thing to say. If “you're told that at least one of them is a girl” was intended to mean “you ask whether at least one of them is a girl and you’re told that at least one of them is a girl” it was pretty easy to make it clear. In that case it’s straightforward that p(you're told at least one is a girl | both are girls) = p(you're told at least one is a girl | only one is a girl) = 1 (with just the mild and reasonable assumption that you’re not being lied to.) Edit: also the reasonable assumption that your question is unrelated to the number of girls - it’s easy to imagine settings where you asked because of something which may be correlated with the number of girls.
AnotherGoodName 1y ago
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.
- zaik 1y ago
  I did not mention a lot of assumptions. For example my problem statement also does not answer the question if Monty always gives you a choice to switch or only when you picked in a certain way. There are quite a few possible variations, all of which change the answer: https://en.wikipedia.org/wiki/Monty_Hall_problem#Other_host_behaviors https://en.wikipedia.org/wiki/Monty_Hall_problem#Other_host_... In the derivation of the likelihood the assumptions are clearly stated. While my knowledge on statistics is certainly unsatisfactory, your assessment of the education I had is quite wrong. My final exam in order to obtain a Master's degree in mathematics is next week. One of the subjects I chose is Bayesian Statistics, so wish me luck.

rtrgrd 1y ago

Might be hug of death but the load times are horrifically slow.

fuomag9 1y ago
For me nextcloud has always been worryingly slow even on my instance
- FabHK 1y ago
  Is the markdown rendered once on the server and stored as HTML? Then why is it slow? Or is it rendered for each client, or rendered in the client?
deleted 1y ago
[deleted]

Shorel 1y ago

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_the_media_furor https://en.wikipedia.org/wiki/Monty_Hall_problem#Savant_and_...

FabHK 1y ago
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.
- kgwgk 1y ago
  > 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://marilynvossavant.com/game-show-problem/ https://web.archive.org/web/20130121183432/http://marilynvos... Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors? [Craig F. Whitaker - Columbia, Maryland] Yes; you should switch. The first door has a 1/3 chance of winning, but the second door has a 2/3 chance. Here’s a good way to visualize what happened. Suppose there are a million doors, and you pick door #1. Then the host, who knows what’s behind the doors and will always avoid the one with the prize, opens them all except door #777,777. You’d switch to that door pretty fast, wouldn’t you?
  - FabHK 1y ago

sparsely 1y ago

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.

zaik 1y ago
If you accept the Bayes theorem, the answer is that the likelihood of "At least one boy is born on a Tuesday" is not the same for different numbers of boys. The more boys the more likely the statement is true. Therefore this information is indicative of how many boys Mrs. Chance has.
- jammaloo 1y ago
  It seems to me that it comes down to how the day of the week was picked. If they picked a random day of the week, and there was only one boy, then there is only a 1/7 chance of a boy being born on that day. If they have one boy, who was born on a Tuesday, and that is why they picked the day, then there is a 100% chance of a boy being born on that day, so no additional information is conferred.
- deleted 1y ago
  [deleted]
- fenomas 1y ago
  Puzzles like that one have always seemed dishonest to me. It only makes sense if you start from the conclusion that it's meant to illustrate Bayes rule, and then work backwards to the assumption that the predicate "boy born on a Tuesday" is supposed to be independent of who's being asked about. But in plain English, ".. At least one of them is a boy born on Tuesday" suggests the speaker is giving a fact that was chosen because it's true of the person spoken about - like if the kids were both chosen on Thursday then that would be the day named. And read that way, the Bayes illustration doesn't stand and the "correct" answer makes no sense. To make it honest, it should really be worded like: "Mrs. Chance has two children of different ages. You ask whether at least one of them is a boy born on Tuesday, and you are told yes. What is the probability that both of them are boys?" Or am I missing something?

qwertytyyuu 1y ago

I feel this the case because pretty much all probability riddles are bayes rules problems which is famously not intuitive

0xfaded 1y ago

Just a tidbit for remembering bayes rule: P(A|B)P(B) = P(A,B) = P(B|A)P(A) The familiar forms P(A|B) = P(A,B)/P(B) = P(B|A)P(A)/P(B) immediately follow

glitchc 1y ago
Comma is not a defined operator. Do you mean intersection?
- 0xfaded 1y ago
  It's a joint distribution :) https://en.m.wikipedia.org/wiki/Joint_probability_distribution https://en.m.wikipedia.org/wiki/Joint_probability_distributi...
bobbylarrybobby 1y ago
In words, for those who need help with the first line: the chance that A and B both happen is the chance that A happens, and (given that A happened) B happens. (Or vice versa.)