Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Nobody Understands Probability

jsteinhardt.wordpress.com

Post reply on HN
Print view View original

21–30 of 64 posts

Re: Nobody Understands Probability

#21

Post by [deleted] »

thwarted wrote: post #18
RickHull wrote: post #16

Earlier quoted context omitted.

> The set of possibilities for two genders of two children is GG, GB, and BB. Yes, but there are two equally probable paths to arrive at (set-theoretic) GB . Each of these paths is equally probably to the remaining paths, BB and GG . There are 4 possible paths, and (set-theoretic) GB is the result of 2 of them. Your application of set theory is inappropriate given 2 independent events.

The question isn't what is the probability of any one of the paths, the question is about the probability of the final result. A fork in the road that joins up again gives each fork equal probability of reaching the destination.

[deleted]

Re: Nobody Understands Probability

#22

Post by mcn »

iliketosleep wrote: post #17
tel wrote: post #15

Earlier quoted context omitted.

The basic formation (which the author argues is not subtle enough to be true) is better thought of step by step. Suppose a man comes up to you and says "I have two children" At this point you build a set of possible realities, your model. There are four possibilities: {BB, BG, GB, GG}. This space fully describes a model whereupon there are two distinct, children with genders. Additionally, via assumption of independe…

Thanks for your explanation. I think his first renormalization process is wrong. Because once we know there is a boy, the problem space is reduced do "What's the probability of a boy?" which is 1/2. It has nothing to do with probabilities involving the known child.

I think the point is that it depends on what prompt the man is implicitly answering.

If you ask parents of two children if they have a boy 3/4 of them will say they do, but if you ask the same parents to randomly name the gender of one of their children 1/2 of them will say they have a boy. Those two different scenarios lead to the 1/3 and 1/2 odds respectively.

Re: Nobody Understands Probability

#23

Post by iliketosleep »

aphyr wrote: post #20
iliketosleep wrote: post #17

Earlier quoted context omitted.

Thanks for your explanation. I think his first renormalization process is wrong. Because once we know there is a boy, the problem space is reduced do "What's the probability of a boy?" which is 1/2. It has nothing to do with probabilities involving the known child.

The problem space is not reduced to the gender of the unspecified child. The important distinction is that "One of my children is a boy" is a statement about both children, not just one of them. Compare that statement to "My first-born child is a boy," and it may make more sense.

I appreciate your reply. But I'm still not clear on it. Because of this: "At least one of them is a boy." As I see it, this statement contains the following pieces of information: 1. There are two children. 2. One of them is a boy. The question is... what's the probability of there being two boys? Considering the information we've got, there are two possible scenarios remaining 1. [B, B] 2. [B, G]. So we have P = 0.5. No?

Re: Nobody Understands Probability

#24

Post by tel »

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.

Obligatory quote: "Essentially, all models are wrong, but some are useful"

- George Box

Re: Nobody Understands Probability

#25

Post by jmtulloss »

If you're lucky enough to attend UIUC, I highly recommend taking ECE 413 to get a thorough introduction to these concepts. It's unfortunate that none of the class materials are online since it goes well above and beyond what is taught in most undergraduate CS courses on statistics. Taking it was hard, but it made me a much better engineer.

Edit: I suppose this applies to anybody in college. Take the hard statistics course that goes over this stuff. It's really valuable, and pretty hard to pick up on your own.

Re: Nobody Understands Probability

#27

Post by seabee »

thwarted wrote: post #18
RickHull wrote: post #16

Earlier quoted context omitted.

> The set of possibilities for two genders of two children is GG, GB, and BB. Yes, but there are two equally probable paths to arrive at (set-theoretic) GB . Each of these paths is equally probably to the remaining paths, BB and GG . There are 4 possible paths, and (set-theoretic) GB is the result of 2 of them. Your application of set theory is inappropriate given 2 independent events.

The question isn't what is the probability of any one of the paths, the question is about the probability of the final result. A fork in the road that joins up again gives each fork equal probability of reaching the destination.

You can't understand the final result in isolation. Out of all men with two children, the probability that they will have two daughters is 1/4, the probability that they will have two sons is 1/4. This leaves heterogeneous offspring occurring 1/2 the time.

If the man is able to make the statement "I have two kids, at least one's a boy", this puts him among the 3/4 of all men with one or two sons. The probability of a man with two sons cannot jump from 1/4 to 3/8 (half of 3/4), as you assert earlier.

It's unintuitive, but it's more obvious when you negate the statement: "I have two children, but I do not have two daughters."

Re: Nobody Understands Probability

#28

Post by ced »

iliketosleep wrote: post #23
aphyr wrote: post #20

Earlier quoted context omitted.

The problem space is not reduced to the gender of the unspecified child. The important distinction is that "One of my children is a boy" is a statement about both children, not just one of them. Compare that statement to "My first-born child is a boy," and it may make more sense.

I appreciate your reply. But I'm still not clear on it. Because of this: "At least one of them is a boy." As I see it, this statement contains the following pieces of information: 1. There are two children. 2. One of them is a boy. The question is... what's the probability of there being two boys? Considering the information we've got, there are two possible scenarios remaining 1. [B, B] 2. [B, G]. So we have P = 0.5…

Three scenarios remain: [B,B], [B,G], [G,B], so the answer is 1/3.

Maybe you were thinking that the order doesn't matter. In that case, what was the probability of getting a boy and a girl, in any order? 1/4 + 1/4 = 1/2. So that's still twice as likely as getting two boys, and that ratio (2:1) will still hold after eliminating [G,G]. You again get 1/3.

Some people find it easier to picture it in terms of frequencies. Imagine 1000 families. What fraction of them have two boys, among those that have at least one boy?

Re: Nobody Understands Probability

#29

Post by kmod »

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…

You unfortunately fall right into the trap that he's warning about: just because you can enumerate the possible outcomes doesn't mean that they are all equally likely (cf: the Monty Hall problem).

The conclusion that BB/BG/GB are all equally likely follows from the assumption that the man would definitely state that he has a male child if and only if one of those conditions were true. But what if instead we add the fact that the man would only say that he has a son if he has no daughters? This isn't contradictory with anything else in the problem, but now the answer is clearly "P(BB) = 1", which makes it hard to state that certainly "P(BB) = 1/3" when there is a consistent interpretation of the problem that gives a different answer.

You should reread the way he set up the problem again: he makes the subtle distinction between things that are given to us by the omniscient problem writer and things that are given to us by characters inside the problem, for whom we have to apply Bayes Law an additional time.

Post reply on HN