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Paradoxes of Probability and Other Statistical Strangeness

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Re: Paradoxes of Probability and Other Statistical Strangeness

#31

Post by dorgo »

marcosdumay wrote: post #29
Houshalter wrote: post #20

Earlier quoted context omitted.

The paradox is that, after picking a random number, you have just done a thing which has probability zero. Doing a thing that has zero probability shouldn't be possible. Ever.

Non-countable sets defy intuition on several ways. The silver lining is that we don't have any evidence a non-countable thing exist on the real world. I don't think anybody even has a procedure for gathering that kind of evidence.

No. Rational number are countable, but have the same property of zero probability for an item in a uniform distribution.

Re: Paradoxes of Probability and Other Statistical Strangeness

#32

Post by Chinjut »

justinpombrio wrote: post #28
Houshalter wrote: post #19

By far the most unintuitive paradox for me personally is the one presented here: https://youtu.be/go3xtDdsNQM?t=3m27s "Mr. Jones has 2 children. What is the probability he has a girl if he has a boy born on Tuesday?" Somehow knowing the day of the week the boy was born changes the result. It's completely bizarre.

The question is ill-posed: it does not give you enough information to tell the probability. You know what Mr. Jones has told you, but you don't know under what circumstances he would have told you this. Suppose that you ask Mr. Jones weather he has a boy and he says yes. Then the probability that he also has a girl is 2/3. Suppose that you asked Mr. Jones weather he had a boy born on a Tuesday, and he says yes. Then…

Everything you say after your first paragraph is correct (presuming people always answer questions with "Yes" or "No" honestly), but…

No one said anything about "Mr. Jones has told you…", here. There was nothing about asking Mr. Jones a question and him providing an answer according to some process.

Rather, the question was simply "Mr. Jones has two children. What is the probability he has a girl if he has a boy born on Tuesday?".

There are implicit conventions involved in reading this, but not particularly problematic ones. This implicitly means "Out of all families with two children, at least one of which is a boy born on Tuesday, what proportion have a girl? [Presuming that out of those families, birth gender and day of the week for the two children are all independently uniformly distributed]". And this is a straightforward counting problem.

So the wording seems fine and the problem well-posed to me.

Re: Paradoxes of Probability and Other Statistical Strangeness

#33

Post by Retric »

Chinjut wrote: post #32
justinpombrio wrote: post #28

Earlier quoted context omitted.

The question is ill-posed: it does not give you enough information to tell the probability. You know what Mr. Jones has told you, but you don't know under what circumstances he would have told you this. Suppose that you ask Mr. Jones weather he has a boy and he says yes. Then the probability that he also has a girl is 2/3. Suppose that you asked Mr. Jones weather he had a boy born on a Tuesday, and he says yes. Then…

Everything you say after your first paragraph is correct (presuming people always answer questions with "Yes" or "No" honestly), but… No one said anything about "Mr. Jones has told you…", here. There was nothing about asking Mr. Jones a question and him providing an answer according to some process. Rather, the question was simply "Mr. Jones has two children. What is the probability he has a girl if he has a boy born…

There are implicit conventions involved in reading this

Explicitly the question adds no such limits. So, abstractly someone could be asking the question without those limits.

It's like the difference between infinity and how whatever subset of math you work in defines infinity. And yes there are more than one commonly used definition.

Re: Paradoxes of Probability and Other Statistical Strangeness

#34

Post by Chinjut »

Retric wrote: post #33
Chinjut wrote: post #32

Earlier quoted context omitted.

Everything you say after your first paragraph is correct (presuming people always answer questions with "Yes" or "No" honestly), but… No one said anything about "Mr. Jones has told you…", here. There was nothing about asking Mr. Jones a question and him providing an answer according to some process. Rather, the question was simply "Mr. Jones has two children. What is the probability he has a girl if he has a boy born…

There are implicit conventions involved in reading this Explicitly the question adds no such limits. So, abstractly someone could be asking the question without those limits. It's like the difference between infinity and how whatever subset of math you work in defines infinity. And yes there are more than one commonly used definition.

Sure, and if the quibble was along the lines of "You never explicitly said boys and girls are 50-50 distributed! You never explicitly said elder and younger childrens' birth genders are independent! You never explicitly said birth-days-of-the-week are uniformly…", then that would be fair, if pedantic.

But this "You know what Mr. Jones has told you, but you don't know under what circumstances he would have told you this" objection is objecting to some other problem than the one posed; the problem posed had nothing to do with Mr. Jones saying anything.

I understand the reason for worrying about this, because many probability riddles ARE poorly worded or presented in such a way as that this becomes an issue, but it wasn't the case here. (Note: I haven't watched rest of the video and have no comment on it; I'm just considering the wording of this individual question within it)

There was never any claim that Mr. Jones said anything, and no one was called to infer anything from any actions taken by Mr. Jones. He could be a lifelong mute. Rather, the fact that Mr. Jones has two children was presented, by an omniscient narrator, and then a counting question was asked.

(Indeed, Mr. Jones himself is completely irrelevant to the problem asked, except as a way of framing the counting question to be about two-children families. The question asked might as well have been "What proportion of two-children families with a boy born on Tuesday have girls?". It was very slightly differently worded, but not in such a way as makes "We don't know what Mr. Jones was asked!" a relevant objection.)

Re: Paradoxes of Probability and Other Statistical Strangeness

#35

Post by cdavid »

philipov wrote: post #17
cdavid wrote: post #15

Earlier quoted context omitted.

That's a different statement: OP is alluding to the fact the measure of Q is 0 when using the "standard" sigma algebra on the real line, while you are saying that the measure of a number of 0. [edit] strictly speaking, you would restrict yourself to a bounded interval, e.g. if you pick a random number from a uniform distribution on [0, 1], the probability that this number is rational is 0.

oh, yeah, but that's because although Q is dense, it is not a dense subset of R and locally that's equivalent to saying a single point is not dense in R

no, it is not. The OP statement is about the probability of the event "{X in Q}" (equal to 0), with X a random variable uniformly distributed on a bounded interval. That even contains many points (infinitely many actually), but has a probability 0.

You are talking about the probability of a single point event, which is also always 0 on that same sigma algebra.

The OP point is not completely trivial because the event contains an infinite (but countably) number of elements. It is fairly easy to understand though since by its very definition, the P[{X in Q}] = sum P[{x}] taken over every rational number (since Q is countable), and each P[{x}] is 0.

A deeper statement is that there exists uncountable sets of probability 0.

Re: Paradoxes of Probability and Other Statistical Strangeness

#36

Post by SomeStupidPoint »

dragonwriter wrote: post #23
georgewsinger wrote: post #6

Earlier quoted context omitted.

Usually in math we assume the axiom of choice :) https://en.wikipedia.org/wiki/Axiom_of_choice I'm assuming this could somehow lead to such a "random" pick in the technical sense. In terms of implementation, I'm not aware of an algorithm that can randomly pick a real number on an actual computer. Perhaps a mathematician could show how to pick one on some abstract machine with infinite resources, and not constrained b…

> In terms of implementation, I'm not aware of an algorithm that can randomly pick a real number on an actual computer An actual (finite in time and space) computer can't even represent arbitrary real numbers, much less randomly choose them.

A Turing machine can't pick random numbers of any kind.

Once you accept that you have an entropy source in the physical world, you can easily be injecting random real numbers (from some range) and in fact, usually are, which are then being binned into integers by ADCs.

Re: Paradoxes of Probability and Other Statistical Strangeness

#37

Post by [deleted] »

Chinjut wrote: post #32
justinpombrio wrote: post #28

Earlier quoted context omitted.

The question is ill-posed: it does not give you enough information to tell the probability. You know what Mr. Jones has told you, but you don't know under what circumstances he would have told you this. Suppose that you ask Mr. Jones weather he has a boy and he says yes. Then the probability that he also has a girl is 2/3. Suppose that you asked Mr. Jones weather he had a boy born on a Tuesday, and he says yes. Then…

Everything you say after your first paragraph is correct (presuming people always answer questions with "Yes" or "No" honestly), but… No one said anything about "Mr. Jones has told you…", here. There was nothing about asking Mr. Jones a question and him providing an answer according to some process. Rather, the question was simply "Mr. Jones has two children. What is the probability he has a girl if he has a boy born…

[deleted]

Re: Paradoxes of Probability and Other Statistical Strangeness

#38

Post by dorgo »

dragonwriter wrote: post #23
georgewsinger wrote: post #6

Earlier quoted context omitted.

Usually in math we assume the axiom of choice :) https://en.wikipedia.org/wiki/Axiom_of_choice I'm assuming this could somehow lead to such a "random" pick in the technical sense. In terms of implementation, I'm not aware of an algorithm that can randomly pick a real number on an actual computer. Perhaps a mathematician could show how to pick one on some abstract machine with infinite resources, and not constrained b…

> In terms of implementation, I'm not aware of an algorithm that can randomly pick a real number on an actual computer An actual (finite in time and space) computer can't even represent arbitrary real numbers, much less randomly choose them.

What about PI? We can represent it in terms of "we know what we are talking about" and we can distinguish it from other numbers.

Re: Paradoxes of Probability and Other Statistical Strangeness

#39

Post by justinpombrio »

Chinjut wrote: post #34
Retric wrote: post #33

Earlier quoted context omitted.

There are implicit conventions involved in reading this Explicitly the question adds no such limits. So, abstractly someone could be asking the question without those limits. It's like the difference between infinity and how whatever subset of math you work in defines infinity. And yes there are more than one commonly used definition.

Sure, and if the quibble was along the lines of "You never explicitly said boys and girls are 50-50 distributed! You never explicitly said elder and younger childrens' birth genders are independent! You never explicitly said birth-days-of-the-week are uniformly…", then that would be fair, if pedantic. But this "You know what Mr. Jones has told you, but you don't know under what circumstances he would have told you th…

> Rather, the fact that he had two children was presented, by an omniscient narrator.

That the narrator is omniscient doesn't change anything. The question still remains: under what circumstances would the narrator have told you, e.g., that "he has a boy born on Tuesday" vs. "he has a girl born on Tuesday". Perhaps this omniscient narrator really likes girls, in which case they would tell you about a girl if Mr. Jones had any girls. Then since they told you "Mr. Jones has a boy born on Tuesday", you know definitely that Mr. Jones has no girls.

Ignoring the source of your knowledge doesn't make that source any less important. And the standard convention you're talking about corresponds to a source of knowledge where you ask a yes/no question and get a yes, which is frequently unrealistic. This is why it disagrees with people's intuition, and this problem is called a paradox.

Re: Paradoxes of Probability and Other Statistical Strangeness

#40

Post by jwilk »

justinpombrio wrote: post #28
Houshalter wrote: post #19

By far the most unintuitive paradox for me personally is the one presented here: https://youtu.be/go3xtDdsNQM?t=3m27s "Mr. Jones has 2 children. What is the probability he has a girl if he has a boy born on Tuesday?" Somehow knowing the day of the week the boy was born changes the result. It's completely bizarre.

The question is ill-posed: it does not give you enough information to tell the probability. You know what Mr. Jones has told you, but you don't know under what circumstances he would have told you this. Suppose that you ask Mr. Jones weather he has a boy and he says yes. Then the probability that he also has a girl is 2/3. Suppose that you asked Mr. Jones weather he had a boy born on a Tuesday, and he says yes. Then…

Following classical probability arguments, we consider a large urn containing two children.

:-O

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