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Paradoxes of Probability and Other Statistical Strangeness

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Re: Paradoxes of Probability and Other Statistical Strangeness

#51

Post by justinpombrio »

Chinjut wrote: post #47
justinpombrio wrote: post #45

Earlier quoted context omitted.

> 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."

It's the fact that it differs from 2/3. If the day of the week was not mentioned the (conventional) answer is exactly 2/3. Not 1/2.

Re: Paradoxes of Probability and Other Statistical Strangeness

#52

Post by oh_sigh »

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.

Is it probability actually zero, or just infinitely close to zero?

Re: Paradoxes of Probability and Other Statistical Strangeness

#53

Post by ubernostrum »

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…

The problem really is almost one of metagaming.

People who are reading this are likely to have seen, for example, questions which read as if they're asking for a conditional probability ("John is male, 33 years old, and has a degree in English literature; what is the probability he works as a barista?") but are designed to let the questioner turn around and say "Ah-HA! I got you! It was really a question about the base rate (in this case, of baristas)!".

As posed and with knowledge of that issue, this question reads like an attempt to do the opposite: to pose a question which seems like it's asking about the base rate of boys vs. girls, but then the questioner turns around with "Ah-HA! I got you! It was really a question about the conditional probability!"

Once it's phrased in a way that makes explicit that it really is a question about conditional probability, and not an attempt to lure someone into a base-rate trap, there's no paradox.

Complicating things is that analyses usually focus on the day of the week as the crucial factor, when it's easier to get to an intuitive understanding of the probability via dealing with the day-of-week first and then focusing on the small but crucial change that comes from knowing the gender of one of the children. After accounting for day-of-week you are left with 28 equally-probable situations, with at least one girl in 14 of them, for the expected 1/2. Then the fact that you end up at a probability just over 1/2 is due to the elimination of the case in which both children are girls (since we know at least one is a boy), which pushes the final result to 13/27 in favor of the second child being a boy.

Re: Paradoxes of Probability and Other Statistical Strangeness

#54

Post by waqf »

bjl wrote: post #50
rocqua wrote: post #46

Earlier quoted context omitted.

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.

Really? Go on, then, pick one and tell us what it is (or at least tell us what your procedure was).

Re: Paradoxes of Probability and Other Statistical Strangeness

#55

Post by waqf »

oh_sigh wrote: post #52
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.

Is it probability actually zero, or just infinitely close to zero?

If it's a random real then the probability is zero. If it's an arbitrarily close approximation to a random real then the probability is arbitrarily close to zero.

Re: Paradoxes of Probability and Other Statistical Strangeness

#56

Post by waqf »

cdavid wrote: post #18
georgewsinger wrote: post #3

Another "paradox": even though it's possible to randomly pick a rational number from the reals, the probability of this happening is 0.

I find that result fairly intuitive, when you understand how measure theory came up to be. A much more surprising result is that most irrational are normal numbers, but we know almost no normal number (morally speaking, a normal number is an irrational number where each digit is equiprobable in any base).

Define "almost no". I mean, they have full measure and I can give countably many explicit examples, what more could you expect?

Re: Paradoxes of Probability and Other Statistical Strangeness

#57

Post by yiyus »

noam87 wrote: post #49
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…

I do not think this is a particularly difficult concept to explain to anyone. A typical example (not difficult to find) and a simple graph are usually good enough for most people.

Re: Paradoxes of Probability and Other Statistical Strangeness

#58

Post by [deleted] »

Shendare wrote: post #44
Chinjut wrote: post #27

Earlier quoted context omitted.

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 referrin…

[deleted]

Re: Paradoxes of Probability and Other Statistical Strangeness

#59

Post by boreas »

For those who might be interested, and in a slightly different vein than the examples in the article, there's the "sleeping beauty" paradox: https://en.wikipedia.org/wiki/Sleeping_Beauty_problem

Basically, an agent is put to sleep and told they will be woken up once or twice, depending on the results of a fair coin flip, without the ability to remember other awakenings.

What probability does the agent assign to the event that the coin landed heads?

The intuitive response is 1/3, but this poses obvious epistemological problems. The agent has, ostensibly, no new information at all, and their prior is surely 1/2. Hope someone else finds this as interesting as I do!

Re: Paradoxes of Probability and Other Statistical Strangeness

#60

Post by astrocat »

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.

I'm an idiot, but I'm going to throw my hat in the ring here:

The video is wrong. The problem reads: Jones has 2 kids. What is P(he has a girl) given that he has a boy born on a Tuesday. Consider, for a moment, what information we're getting from "boy born on a Tuesday." This is no different than "boy with red hair," or "boy with 5 freckles." The fact that the BOY was born on a tuesday does not change P(day of the week girl was born). Imagine the "boy with 5 freckles" case - let 5 freckles be denoted by F5, six freckles by F6 and so on... would the appropriate calculation include enumerating P(boy F5, boy Fn) for all n? No.

The "born on Tuesday" is irrelevant. Thus you have the following scenarios: - one kid is TuesdayBoy and the other is also a boy, born at any time - one kid is TuesdayBoy and the other is a girl, born at any time

Out of these options P(Jones has a girl) is a flat out 50%. There is no need to bring in concepts of "which was born first" or enumerate all possible days of the week each child could have been born.

Ok... now all the real smartypants here can correct me :)

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