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Paradoxes of Probability and Other Statistical Strangeness

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Re: Paradoxes of Probability and Other Statistical Strangeness

#71

Post by ouid »

colonelxc wrote: post #64
boreas wrote: post #59

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 e…

I mostly find it interesting in that people could think that the chance is 1/3 (and that it may even be obvious!). After reading the description I can understand what they are getting at, but I think the conditional probability is messed up. Instead of P(Monday | Heads) = P(Monday | Tails) = P(Tuesday | Tails) it is really P(Monday | Heads&Awake) = P(Monday | Tails&Awake) = P(Tuesday| Tails&Awake) or something like t…

You will be awoken twice as many times because the coin comes up tails as you will because the coin comes up heads. If you can manipulate the formulae to tell you something different, that only means you have failed to manipulate the formulae correctly.

Re: Paradoxes of Probability and Other Statistical Strangeness

#72

Post by ouid »

waqf wrote: post #56
cdavid wrote: post #18

Earlier quoted context omitted.

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?

I think all mathematicians, given enough time, will eventually say countable to mean infinite.

Re: Paradoxes of Probability and Other Statistical Strangeness

#73

Post by ouid »

stiff wrote: post #7
Sinergy2 wrote: post #4

Earlier quoted context omitted.

Please describe how it is possible to pick such a number. For example, I can readily imagine how to pick a random 32b float, but that it is an entirely problem with a nonzero probability.

In probability theory, when dealing with continuous sample spaces / random variables, events with probability 0 still have a chance of occurring, and events with probability 1 stil l have a chance of NOT occurring, see: https://en.wikipedia.org/wiki/Almost_surely This strange property comes from strange properties of the real numbers (and uncountably infinite sets) that give rise to things like: https://en.wikipedia.…

What's an anagram for Banach Tarski?

Banach Tarski Banach Tarski.

Seriously though, you can do math without invoking the axiom of choice. The formulation of probability doesn't strictly depend on it.

Re: Paradoxes of Probability and Other Statistical Strangeness

#74

Post by ouid »

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.

Those things sound equivalent, but I don't want to be the one to try to prove it.

Re: Paradoxes of Probability and Other Statistical Strangeness

#75

Post by roenxi »

colonelxc wrote: post #64
boreas wrote: post #59

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 e…

I mostly find it interesting in that people could think that the chance is 1/3 (and that it may even be obvious!). After reading the description I can understand what they are getting at, but I think the conditional probability is messed up. Instead of P(Monday | Heads) = P(Monday | Tails) = P(Tuesday | Tails) it is really P(Monday | Heads&Awake) = P(Monday | Tails&Awake) = P(Tuesday| Tails&Awake) or something like t…

The outcomes can be easily enumerated. If Sleeping Beauty always answers "Heads" she will be right 33% of the times she is asked. This is pretty close to the definition of a 33% chance.

She hasn't been given new information by waking up, she also knows as she goes into the experiment - "most of the outcomes where I am being interviewed involve the coin toss coming up tails".

Re: Paradoxes of Probability and Other Statistical Strangeness

#76

Post by bo1024 »

boreas wrote: post #59

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 e…

It's very interesting and I don't think there's an obvious correct answer. It's hard to formally model mathematically.

Here's a game-theoretic perspective. In general, when an event has a 1/3 chance of happening, an idealized gambler would be indifferent between the following two bets or lottery tickets: (A) win $2 if the event happens; (B) win $1 if the event doesn't happen. (Notice her average payoff is 2/3 no matter which bet she takes.)

Now in the sleeping beauty problem where tails is two awakenings and heads is one, a gambler would be indifferent between (A) winning $2 every time she wakes up and the coin is heads, and (B) winning $1 every time she wakes up and the coin is tails. This suggests that her "belief" is 1/3.

Another way to put it might be that for a risk-neutral agent, doubling the payoff in one state of the world is equivalent to doubling its "perceived probability". In the sleeping beauty problem, doubling the payoff is like experiencing everything twice.

Re: Paradoxes of Probability and Other Statistical Strangeness

#77

Post by Chinjut »

justinpombrio wrote: post #51
Chinjut wrote: post #47

Earlier quoted context omitted.

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.

Yes, that's also "paradoxical", though probably not the paradox that would trip people up first unless they'd seen the other problem first. But, you're right that I may have misread which departure from expected answer was bugging the OP. Nonetheless, everything else I stated still holds.

Re: Paradoxes of Probability and Other Statistical Strangeness

#78

Post by Chinjut »

adverbly wrote: post #61
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…

I also fell into the same ambiguity trap, and I think that the objection about explicit wording is a fair one to make. "What proportion of two-children families with a boy born on Tuesday have girls?" seems completely clear to me. I would have answered that question relatively quickly. But the original question had me very confused. I felt a strong desire to ask more about the situation. A great deal of my intuition…

I mean, it's just as arbitrary in my rephrased version. I could just as well ask "What proportion of two-children families with a boy born on Monday have girls?". But, very well, the different wording prompted differing intuitions for you; so it goes.

Re: Paradoxes of Probability and Other Statistical Strangeness

#79

Post by Chinjut »

astrocat wrote: post #60
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 we…

There are 2 * 7 * 2 * 7 ways to assign gender and birth-day-of-week to two children. By convention, all are considered equiprobable (this is the same as assuming kids' genders and birth day-of-weeks are independent of each other and of all facts about other kids, and that both genders are equally likely and all 7 days are equally likely for any given kid.)

Of these possibilities, 27 are situations where one kid is a Tuesday boy. [Do you dispute this count?]

Of those, 14 are situations where one kid is a girl. [Do you dispute this count?]

The answer to "What proportion of cases where there is at least one Tuesday boy also have a girl?" is thus 14/27.

You have stated by fiat that certain things are irrelevant to certain other things, that certain things have probability 50%, etc, but in doing so, you have not considered the count correctly. You are likely misled by phrasing such as "the boy", when there are families with two boys in which there is no proper referent of "the boy" and no particular answer to question like "Which day was 'the boy' born?".

Re: Paradoxes of Probability and Other Statistical Strangeness

#80

Post by ubernostrum »

astrocat wrote: post #60
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 we…

First step back and consider the possibilities given no knowledge whatsoever:

For each child the problem constrains to one of two possible sexes and one of seven possible days of birth.

2 * 7 = 14 possible sex/day combinations for a single child.

(2 * 7) * (2 * 7) = 196 possible sex/day combinations for a pairing of two children. To see why, you could write a program to enumerate all of them, starting with the pairing "Boy/Monday + Boy/Monday", then "Boy/Monday + Boy/Tuesday" and so on until you exhaust all possible options at "Girl/Sunday + Girl/Sunday". You'll see there are 196 options.

Now start applying the facts given to us: one of the children is born on a Tuesday (eliminate all possibilities which don't have at least one Tuesday child), and that child is a boy (eliminate all possibilities in which there is not a Tuesday child who is also a boy).

This leaves exactly 27 possible cases:

Boy/Sunday + Boy/Tuesday,

Boy/Monday + Boy/Tuesday,

Boy/Tuesday + Boy/Tuesday,

Boy/Wednesday + Boy/Tuesday,

Boy/Thursday + Boy/Tuesday,

Boy/Friday + Boy/Tuesday,

Boy/Saturday + Boy/Tuesday,

Girl/Sunday + Boy/Tuesday,

Girl/Monday + Boy/Tuesday,

Girl/Tuesday + Boy/Tuesday,

Girl/Wednesday + Boy/Tuesday,

Girl/Thursday + Boy/Tuesday,

Girl/Friday + Boy/Tuesday,

Girl/Saturday + Boy/Tuesday,

Boy/Tuesday + Boy/Sunday,

Boy/Tuesday + Boy/Monday,

Boy/Tuesday + Boy/Wednesday,

Boy/Tuesday + Boy/Thursday,

Boy/Tuesday + Boy/Friday,

Boy/Tuesday + Boy/Saturday,

Boy/Tuesday + Girl/Sunday,

Boy/Tuesday + Girl/Monday,

Boy/Tuesday + Girl/Tuesday,

Boy/Tuesday + Girl/Wednesday,

Boy/Tuesday + Girl/Thursday,

Boy/Tuesday + Girl/Friday,

Boy/Tuesday + Girl/Saturday

If you count, you'll see that of those 27, there are 13 with two boys and 14 with a boy and a girl. The probability of two boys, given that one child is a boy born on Tuesday, is thus 13/27.

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