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Nobody Understands Probability

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Re: Nobody Understands Probability

#41

Post by Tichy »

thwarted wrote: post #13
RickHull wrote: post #8

> However, the answer is not, in fact, 1/3. Why is this? This seems like a canard to me. Here is my defense of 1/3 as a correct answer: http://gist.github.com/578386 > Is Bayes’ theorem wrong? > No, the answer comes from an unfortunate namespace collision in the word “given”. The man “gave” us the information that he has at least one male child. By this we mean that he asserted the statement “I have at least one male…

1/3 isn't correct, and I find the OP's explanation to be overly complex. The set of possibilities for two genders of two children is GG, GB, and BB. In your possibilities, GB and BG are exactly the same set (order doesn't matter in a set, only membership), so you don't have 4 possibilities, you have 3 total. Since the guy asserted that one of them is a boy, you can rule out the GG possibility. This leaves only GB and…

You picked that 0.5 out of thin air, though. You forgot about BG.

Re: Nobody Understands Probability

#42

Post by Tichy »

Dn_Ab wrote: post #38
RickHull wrote: post #8

> However, the answer is not, in fact, 1/3. Why is this? This seems like a canard to me. Here is my defense of 1/3 as a correct answer: http://gist.github.com/578386 > Is Bayes’ theorem wrong? > No, the answer comes from an unfortunate namespace collision in the word “given”. The man “gave” us the information that he has at least one male child. By this we mean that he asserted the statement “I have at least one male…

I thought on this a bit. And I do believe you are correct. Because as you note, he did in fact condition the space such that 1/3 is the correct answer and is stretching. He goes on to say "Now this means that if we want to claim that the probability that the man has two boys is , what we are really claiming is that he is equally likely to inform us that he has at least one boy, in all situations where it is true, ind…

Can't read your code, but aren't (F,M,1) and (M,F,2) impossible? Not sure what exactly you are calculating.

Re: Nobody Understands Probability

#43

Post by Confusion »

tome wrote: post #39
frossie wrote: post #2

For those who didn't make it all the way down: People often intuitively think of probabilities as a fact about the world, when in reality probabilities are a fact about our model of the world.

There are at least two interpretations of probability. Firstly the epistemological, as you say, represents our lack of knowledge about the world. Secondly the aleatory truly represents the phenomenon of chance in the world. The best interpretation of quantum theory for example (as I understand it) takes the latter view that randomness is genuinely physically manifested, and does not simply represent our inability to…

  The best interpretation of quantum theory [..] takes the [..] view that
  randomness is genuinely physically manifested
We could argue endlessly about whether that is 'the best' interpretation (which ethical assumptions does your 'the best' presuppose?) but fortunately it doesn't matter. As Mermin (from the famous Ashcroft and Mermin book on Solid State Physics) famously quipped: you can 'shut up and calculate'. The usefulness of the model does not depend on its interpretations (although the interpretations are certainly important with respect to scientific progress).

Re: Nobody Understands Probability

#44

Post by tome »

Confusion wrote: post #43
tome wrote: post #39

Earlier quoted context omitted.

There are at least two interpretations of probability. Firstly the epistemological, as you say, represents our lack of knowledge about the world. Secondly the aleatory truly represents the phenomenon of chance in the world. The best interpretation of quantum theory for example (as I understand it) takes the latter view that randomness is genuinely physically manifested, and does not simply represent our inability to…

The best interpretation of quantum theory [..] takes the [..] view that randomness is genuinely physically manifested We could argue endlessly about whether that is 'the best' interpretation (which ethical assumptions does your 'the best' presuppose?) but fortunately it doesn't matter. As Mermin (from the famous Ashcroft and Mermin book on Solid State Physics) famously quipped: you can 'shut up and calculate'. The us…

Well indeed, and this doesn't just apply to quantum mechanics but to the OP's quotation from the article too:

People often intuitively think of probabilities as a fact about the world, when in reality probabilities are a fact about our model of the world.

Re: Nobody Understands Probability

#45

Post by scott_s »

tome wrote: post #39
frossie wrote: post #2

For those who didn't make it all the way down: People often intuitively think of probabilities as a fact about the world, when in reality probabilities are a fact about our model of the world.

There are at least two interpretations of probability. Firstly the epistemological, as you say, represents our lack of knowledge about the world. Secondly the aleatory truly represents the phenomenon of chance in the world. The best interpretation of quantum theory for example (as I understand it) takes the latter view that randomness is genuinely physically manifested, and does not simply represent our inability to…

I think you're confusing concepts. The original point is about the difference between the map and the territory. And, because I'm (finally) systematically going through Eliezer's sequences: http://wiki.lesswrong.com/wiki/Map_and_Territory_(sequence)

Your second paragraph is about a particular map: quantum theory. Quantum theory has probabilities in it. The dominant interpretation of quantum theory is that the probabilities accurately represent what happens in the universe; they are not artifacts for us to correct. But there is still a difference between our map (quantum theory) and the territory (the universe itself).

Put another way: quantum theory is a map with uncertainty baked into it. But this uncertainty has been accurately mapped.

Re: Nobody Understands Probability

#46

Post by abeppu »

thwarted wrote: post #13
RickHull wrote: post #8

> However, the answer is not, in fact, 1/3. Why is this? This seems like a canard to me. Here is my defense of 1/3 as a correct answer: http://gist.github.com/578386 > Is Bayes’ theorem wrong? > No, the answer comes from an unfortunate namespace collision in the word “given”. The man “gave” us the information that he has at least one male child. By this we mean that he asserted the statement “I have at least one male…

1/3 isn't correct, and I find the OP's explanation to be overly complex. The set of possibilities for two genders of two children is GG, GB, and BB. In your possibilities, GB and BG are exactly the same set (order doesn't matter in a set, only membership), so you don't have 4 possibilities, you have 3 total. Since the guy asserted that one of them is a boy, you can rule out the GG possibility. This leaves only GB and…

1/3 would be correct if you set aside the issues about 'given' information addressed in the post. And I think the post has a valid point, that we have to be careful to appropriately understand biases and uncertainties in our data sources. But your reading of it, I think, demonstrates the real weakness of this part of the OP's discussion, namely that he's started with, and trying to expand from, a basic, 'classic' problem that people notoriously have difficulty agreeing on (e.g. http://en.wikipedia.org/wiki/Maryln_vos_Savant#.22Two_boys.2...). So as a pedagogical exercise, or as an attempt to get his point across, he has to trip over all the misunderstandings that cause people to think in the simple, classic case, that the answer is 1/2.

I think he would have been better off expanding the classic coin example, only changing the story such that a naive research assistant is tasked with flipping the coin n times and bringing us the results. Then we could start by finding the probability that the coin is biased taking the research assistant at his word, and then we could reason about the probability that the research assistant either doctored the sequence when he felt they weren't 'random' enough, or the probability that he just pulled an 'HTTH' sequence out of his head without ever touching a coin.

Re: Nobody Understands Probability

#47

Post by thwarted »

abeppu wrote: post #46
thwarted wrote: post #13

Earlier quoted context omitted.

1/3 isn't correct, and I find the OP's explanation to be overly complex. The set of possibilities for two genders of two children is GG, GB, and BB. In your possibilities, GB and BG are exactly the same set (order doesn't matter in a set, only membership), so you don't have 4 possibilities, you have 3 total. Since the guy asserted that one of them is a boy, you can rule out the GG possibility. This leaves only GB and…

1/3 would be correct if you set aside the issues about 'given' information addressed in the post. And I think the post has a valid point, that we have to be careful to appropriately understand biases and uncertainties in our data sources. But your reading of it, I think, demonstrates the real weakness of this part of the OP's discussion, namely that he's started with, and trying to expand from, a basic, 'classic' pro…

http://en.wikipedia.org/wiki/Boy_or_Girl_paradox has interesting content in the sections Second Question and Ambiguous Problem Statements.

I'll accept that I'm wrong and that the answer is 1/3 and that I just don't understand it and that I'm supporting the point that "no one understands probability, especially me", but there has been little explanation why the birth order is even a factor. Even that Wikipedia entry labels the four possibilities with "older child" and "younger child", which seems to be extra, unneeded information when the term "at least" is used to describe the number of boys independent of order.

Re: Nobody Understands Probability

#48

Post by thwarted »

Tichy wrote: post #41
thwarted wrote: post #13

Earlier quoted context omitted.

1/3 isn't correct, and I find the OP's explanation to be overly complex. The set of possibilities for two genders of two children is GG, GB, and BB. In your possibilities, GB and BG are exactly the same set (order doesn't matter in a set, only membership), so you don't have 4 possibilities, you have 3 total. Since the guy asserted that one of them is a boy, you can rule out the GG possibility. This leaves only GB and…

You picked that 0.5 out of thin air, though. You forgot about BG.

I didn't "forget" about it, I contend that case is already handled by GB; "one child is a boy and one child is a girl" is the same as "one child is a girl and one child is a boy" and can be expressed as either GB or BG, making GB and BG equivalent.

Re: Nobody Understands Probability

#49

Post by abeppu »

thwarted wrote: post #47
abeppu wrote: post #46

Earlier quoted context omitted.

1/3 would be correct if you set aside the issues about 'given' information addressed in the post. And I think the post has a valid point, that we have to be careful to appropriately understand biases and uncertainties in our data sources. But your reading of it, I think, demonstrates the real weakness of this part of the OP's discussion, namely that he's started with, and trying to expand from, a basic, 'classic' pro…

http://en.wikipedia.org/wiki/Boy_or_Girl_paradox has interesting content in the sections Second Question and Ambiguous Problem Statements. I'll accept that I'm wrong and that the answer is 1/3 and that I just don't understand it and that I'm supporting the point that "no one understands probability, especially me", but there has been little explanation why the birth order is even a factor. Even that Wikipedia entry l…

Birth order per se doesn't matter; probability mass does. Birth order is an easy way to show that you're twice as likely to get a boy and a girl as you are to get two boys. But you can ignore birth order and say instead "let k be a binomially distributed random variable with n=2 and p=1/2. If you know k>=1 what is p(k=2)?". Then the answer is a straight comparison of p(k=2) to p(k=1), where we note that '2 choose 1' is 2, and '2 choose 2' is 1. Note that ordering per se isn't entering into this, but to someone without probability background, it's a bit obtuse. Describing the probability space in terms of birth order is not strictly necessary but helps a lot of people grasp the concept.

Re: Nobody Understands Probability

#50

Post by Tichy »

thwarted wrote: post #48
Tichy wrote: post #41

Earlier quoted context omitted.

You picked that 0.5 out of thin air, though. You forgot about BG.

I didn't "forget" about it, I contend that case is already handled by GB; "one child is a boy and one child is a girl" is the same as "one child is a girl and one child is a boy" and can be expressed as either GB or BG, making GB and BG equivalent.

No, choosing to not care for order changes the way probabilities have to be calculated. I suppose there was enough discussion about that on this thread already.

I guess it is helpful to think about two coin flips instead of boys and girls.

Not sure how it is being taught in the US. In Germany we use urn models and learn formulas for drawing balls from urns in different variations. Like returning the balls or not returning the balls, or caring for order or not. All these cases have different formulas attached...

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