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Paradoxes of Probability and Other Statistical Strangeness

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Re: Paradoxes of Probability and Other Statistical Strangeness

#41

Post by x0x0 »

justinpombrio wrote: post #39
Chinjut wrote: post #34

Earlier quoted context omitted.

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. Jone…

As a probability problem with the standard assumptions, it's a well defined question. If you saw this in Bertsekas or Sheldon Ross, the sampling would be clear.

And I also think you're incorrect about why it's a paradox. People are just bad at understanding and estimating things in conditional probabilities. Further, the answer changes based on the sampling regime, which (as mentioned) was not explicitly stated but is clear to almost any student that's taken a discrete probability class.

Re: Paradoxes of Probability and Other Statistical Strangeness

#42

Post by thousandautumns »

beefield wrote: post #21
fitchjo wrote: post #12

My favorite statistical/probability paradox has always been the birthday paradox.

I don't know if Monty hall problem counts as a paradox, but that is quite high on my favourite list of counterintuitive probability results.

In my experience the only reason the Monty Hall problem comes off as paradoxical is because it is usually poorly explained.

Re: Paradoxes of Probability and Other Statistical Strangeness

#43

Post by leni536 »

dorgo wrote: post #38
dragonwriter wrote: post #23

Earlier quoted context omitted.

> 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.

You can only have countable number of first or second order logic statements each defining a specific real number.

Re: Paradoxes of Probability and Other Statistical Strangeness

#44

Post by Shendare »

Chinjut wrote: post #27
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.

Your problem is that you are thinking there's a "the boy". But there's not a "the boy". Mr. Jones could have two boys. He could have two boys both born on Tuesday, even. The term "the boy" does not denote any particular boy, in that case, and causes you to think about the situation erroneously. If the question were "There's Kid 1 and Kid 2, each independently selected with random gender and birth-day-of-the-week. Out…

Agreed. He removed the second B2B2 probability annotation as though it were a repeat of the first and inapplicable to the probability set, but that's not the case, and it shouldn't be removed. Apply lower-case to the younger boy in the probability sets and it's clear why. B2b2 is not the same occurrence as b2B2. Even though the day both were born on was "a Tuesday" doesn't mean both probability instances are referring to the exact same event. Except in the case of twins, which is outside the scope of the exercise.

Re: Paradoxes of Probability and Other Statistical Strangeness

#45

Post by justinpombrio »

x0x0 wrote: post #41
justinpombrio wrote: post #39

Earlier quoted context omitted.

> 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. Jone…

As a probability problem with the standard assumptions, it's a well defined question. If you saw this in Bertsekas or Sheldon Ross, the sampling would be clear. And I also think you're incorrect about why it's a paradox. People are just bad at understanding and estimating things in conditional probabilities. Further, the answer changes based on the sampling regime, which (as mentioned) was not explicitly stated but i…

> And I also think you're incorrect about why it's a paradox. People are just bad at understanding and estimating things in conditional probabilities.

This is a testable prediction. I predict that making the source of your knowledge explicit eliminates the paradox.

To me, it feels strange that "the probability that Mr. Jones has a girl given that he has a boy born on Tuesday" is ~1/2. However, it feels normal that "You ask Mr. Jones weather he has a boy born on Tuesday, and he says yes. What is the probability that he has a girl?" is ~1/2.

Do other people agree?

Re: Paradoxes of Probability and Other Statistical Strangeness

#46

Post by rocqua »

Houshalter wrote: post #20
philipov wrote: post #13

Earlier quoted context omitted.

I think this only sounds like a paradox if it is phrased poorly. The accurate way to state it is "The probability of randomly picking a specific number is 0" and that sounds reasonable. The probability of successfully picking any number is 1.

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.

You can't pick a random real number between 0 and 1. Heck, almost all reals between 0 and 1 can't ever be constructed let alone picked.

The here is the non-constructive nature of the real numbers. That is not to say the reals are useless, but they are not much more than a formalism. It's rather useful though because it's hard to get numbers like pi or e. Its really nice that any real interval is compact, but that too is hard to replicate.

Re: Paradoxes of Probability and Other Statistical Strangeness

#47

Post by Chinjut »

justinpombrio wrote: post #45
x0x0 wrote: post #41

Earlier quoted context omitted.

As a probability problem with the standard assumptions, it's a well defined question. If you saw this in Bertsekas or Sheldon Ross, the sampling would be clear. And I also think you're incorrect about why it's a paradox. People are just bad at understanding and estimating things in conditional probabilities. Further, the answer changes based on the sampling regime, which (as mentioned) was not explicitly stated but i…

> And I also think you're incorrect about why it's a paradox. People are just bad at understanding and estimating things in conditional probabilities. This is a testable prediction. I predict that making the source of your knowledge explicit eliminates the paradox. To me, it feels strange that "the probability that Mr. Jones has a girl given that he has a boy born on Tuesday" is ~1/2. However, it feels normal that "Y…

It's not that the probability is close to 1/2 that makes it paradoxical for most people. It's that the probability differs from 1/2 at all. As in the OP of this very thread saying "Somehow knowing the day of the week the boy was born changes the result. It's completely bizarre."

Re: Paradoxes of Probability and Other Statistical Strangeness

#48

Post by Chinjut »

Chinjut wrote: post #27
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.

Your problem is that you are thinking there's a "the boy". But there's not a "the boy". Mr. Jones could have two boys. He could have two boys both born on Tuesday, even. The term "the boy" does not denote any particular boy, in that case, and causes you to think about the situation erroneously. If the question were "There's Kid 1 and Kid 2, each independently selected with random gender and birth-day-of-the-week. Out…

The 14/27 answer in the video is correct, incidentally.

Also, I notice you said "Somehow knowing the day of the week the boy was born changes the result. It's completely bizarre."

Remember, though, there's no "the boy". The question "On which day of the week was the boy born? Tell me, I need to know!" does not always have a well-defined answer.

Indeed, you'd get the same 14/27 answer even if "Tuesday" in the question "What proportion of two-children families with at least one Tuesday boy have a girl?" was replaced by any other day. And if this seems paradoxically in conflict with the fact that simply asking "What proportion of two-children families with at least one boy have a girl?" has instead the answer 2/3, reflect again upon the fact that some families have two boys born on different days, so that there's no single answer to "On what day was 'the boy' born?". And then just draw out the cases and count.

(Specifically, out of the 2 * 7 * 2 * 7 equiprobable cases overall for Kid 1 and Kid 2's genders and days, there are 27 cases where there's at least one Tuesday boy, and 14 cases where there's at least one Tuesday boy and also a girl. There are 3 * 7^2 cases where there's at least one boy, and 2 * 7^2 cases where there's at least one boy and also a girl.)

Many of these questions, I think, become clearer if thought of as counting questions instead of as "probability" questions (though it's all the same; the math called "probability" is just the math of various kinds of counting (from simple counting as in this case to complexly weighted continuous measurements, but still ultimately a generalized form of counting). However, despite that equivalence, the concept "probability" has developed all these other distracting connotations, such that psychologically, there can be a useful difference in perspective in switch to explicitly thinking "counting" instead. No one would long dispute that there are 27 cases with at least one Tuesday boy, etc.).

Re: Paradoxes of Probability and Other Statistical Strangeness

#49

Post by noam87 »

fitchjo wrote: post #12

My favorite statistical/probability paradox has always been the birthday paradox.

For me it's Simpson's paradox: it throws everyone off -- it's caused (and will continue to cause) real-world damage, it's everywhere once you see it -- it's in how newspapers report science, it's in our social policy and how we talk about social issues, it's in court cases --, and finally, it's really hard to explain to a non-math person; so even when it's happening, you sound like the irrational one for pointing it out.

... and don't get cocky once you know about it, because it's so pernicious it'll get you too if you're not careful!

Re: Paradoxes of Probability and Other Statistical Strangeness

#50

Post by bjl »

rocqua wrote: post #46
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.

You can't pick a random real number between 0 and 1. Heck, almost all reals between 0 and 1 can't ever be constructed let alone picked. The here is the non-constructive nature of the real numbers. That is not to say the reals are useless, but they are not much more than a formalism. It's rather useful though because it's hard to get numbers like pi or e. Its really nice that any real interval is compact, but that too…

You can certainly pick a random real number from the unit interval.
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