A fun little probability puzzle for you.

[Flannel Jesus]

Okay, if you don't understand words then I guess I can't achieve my goal. Oh well.

[bobmax]

When I have the $ 100 bill in my hand, it is uniquely identified, this is it right here.

It can be of the box of 100 and 1 or of the box of 100 and 100.
In both cases I take it off and it stays in the first box $ 1 and in the other $ 100.

And so I conclude that the probability is 1/2.

Why am I wrong?

In my opinion, because I do not consider that in the second box the 100 bill I have in my hand is not unique, but it is perfectly equivalent to that other one.

The majority of people who treat the 2 100s in the double box as 'not unique' are the ones saying that the probability is 1/2. The people who treat them as unique are the ones saying the probability is 2/3. That's why I'm not jiving with this explanation - I think it's exactly the wrong way around.

When you treat the 2x 100s as unique, you INCREASE the surface area of the probability to be the second box. Treating them as unique is arguably one of the biggest reasons why we say it's 2/3 and not 1/2. When you treat them as identical, rather than unique, you don't count them twice, you count them together once, and that line of reasoning is exactly why a lot of people intuit that the probability is 1/2.

This is the correct thinking, the two bills are both unique.

But what I try to say is that the uniqueness of the ticket in my hand is wrong to the detriment of the other one.
That it could have been in its place without changing anything.

I have the impression that perhaps we are saying the same thing.
I am only describing the intellectual process that led me to answer 1/2. But something was wrong...
I had overloaded the significance of that 100 bill.

[Skepdick]

Okay, if you don't understand words then I guess I can't achieve my goal. Oh well.

You are never going to achieve your circular goal. Not because I don't understand, but because your words are meaningless.

To pass judgment re: under, or over-estimation you need to be explicit about the reference frame and coordinates (in spacetime) for "true value"

[Flannel Jesus]

This is the correct thinking, the two bills are both unique.

But what I try to say is that the uniqueness of the ticket in my hand is wrong to the detriment of the other one.
That it could have been in its place without changing anything.

I have the impression that perhaps we are saying the same thing.
I am only describing the intellectual process that led me to answer 1/2. But something was wrong...
I had overloaded the significance of that 100 bill.

Without changing anything fom the perspective of the experimentee, yes.

But from an analytical perspective, treating them as separate changes everything.

The person who treats the 2 100s as identical analyses the situation like this:

I could have selected box(1+100), and selected a $100
or I could have selected box(100+100) and selected a $100.
50/50 chance.

The person who treats the 2 100s as unique analyses the situation like this:
I could have selected box(1+100), and selected a $100
or I could have selected box(100+100) and selected a $100,
or I could have selected box(100+100) and selected THE OTHER $100,
2/3 chance that I selected the 100+100 box.

That's why the second way is correct. AND, I've verified, the second way generally gives you the correct answer in other similar questions as well.

[Flannel Jesus]

Okay, if you don't understand words then I guess I can't achieve my goal. Oh well.

You are never going to achieve your circular goal. Not because I don't understand, but because your words are meaningless.

To pass judgment re: under, or over-estimation you need to be explicit about the reference frame and coordinates (in spacetime) for "spot-on-estimation".

Do you want me to explain how everybody else means 'over/underestimation' in this context? Or do you want to just stay with your own interpretation and remain ignorant of how everybody else is approaching the conversation?

[Skepdick]

Do you want me to explain how everybody else means 'over/underestimation' in this context? Or do you want to just stay with your own interpretation and remain ignorant of how everybody else is approaching the conversation?

I know how everyone else is using it.

overestimation noun a judgement or rough calculation that is too favourable or too high
underestimation noun a judgement or rough calculation that is unfavourable or too low.

Too high/low with respect to... what? It can't be with respect to an estimation. Estimations either too high; or too low by definition.

What's your baseline?

[bobmax]

The person who treats the 2 100s as unique analyses the situation like this:
I could have selected box(1+100), and selected a $100
or I could have selected box(100+100) and selected a $100,
or I could have selected box(100+100) and selected THE OTHER $100,
2/3 chance that I selected the 100+100 box.

That's why the second way is correct. AND, I've verified, the second way generally gives you the correct answer in other similar questions as well.

Yup!

And this third possibility emerges when we consider that the selected bill could have been that OTHER. Because the two are equivalent.

[Flannel Jesus]

I know how everyone else is using it.

overestimation noun a judgement or rough calculation that is too favourable or too high
underestimation noun a judgement or rough calculation that is unfavourable or too low.

Too high/low with respect to... what? It can't be with respect to an estimation. Estimations either too high; or too low by nature.

Too high, or too low, with respect to the *correct probability calculation*.

NOT "correct" in the way you seem to think. You're obsessed with reducing each option down to a probability of 1/0 by looking in the box in 1 particular run of the experiment. Nobody else cares about that. Nothing could be more boring or uninteresting than that.

Correct analytically, probabilistically.

If I show you a coin and ask you to guess the probability of it landing heads, and you say "50%", and it was actually a weighted coin that lands heads 75% of the time, then 50% is an underestimation compared to the analytically correct probability of 75%.

Many people guess that the probability that you've selected the 100+100 box after seeing the first 100 is 50/50. The correct answer, not in one particular instance but analytically, probabilistically, across all instances, is 2/3. 50% is an underestimate compared to the correct probability calculation.

That's what he means by over/underestimation.

[Flannel Jesus]

The person who treats the 2 100s as unique analyses the situation like this:
I could have selected box(1+100), and selected a $100
or I could have selected box(100+100) and selected a $100,
or I could have selected box(100+100) and selected THE OTHER $100,
2/3 chance that I selected the 100+100 box.

That's why the second way is correct. AND, I've verified, the second way generally gives you the correct answer in other similar questions as well.

Yup!

And this third possibility emerges when we consider that the selected bill could have been that OTHER. Because the two are equivalent.

Haha, this is where we differ on our words here I guess. The people who treat them as fully equivalent are not saying "The other", only people who see them as distinct and unique are recognizing that "the other 100" is even a meaningful thing to say.

[Flannel Jesus]

Perhaps I'm treating 'unique' and 'equivalent' as antonyms here, while you're treating them as borderline synonymous.

But as long as you count each $100 separately, you come to the right answer. We agree on that.

[Skepdick]

Too high, or too low, with respect to the *correct probability calculation*.

Weasel words. Which calculation is the "correct" one? The true probability, or your estimate?

Correct analytically, probabilistically.

Don't you think that judgments of "correctness" about analytical statements are wee bit premature?

It's a bit like marking your own homework.

If I show you a coin and ask you to guess the probability of it landing heads, and you say "50%", and it was actually a weighted coin that lands heads 75% of the time, then 50% is an underestimation compared to the analytically correct probability of 75%.

Exactly! So my calculation wasn't correct! Why? because I (falsely) assumed it was a fair coin!

I started with the wrong prior and I under-estimated the true probability.

Garbage in - Garbage out.

Many people guess that the probability that you've selected the 100+100 box after seeing the first 100 is 50/50. The correct answer, not in one particular instance but analytically, probabilistically, across all instances, is 2/3. 50% is an underestimate compared to the correct probability calculation.

That's what he means by over/underestimation.

So you are, in fact committing the Ludic fallacy.

Just as I thought though... unless you are idea-driven your notion of "correctness" is really tautologous/useless.

[Flannel Jesus]

I don't understand how you can say "correct probability" is weasel words, and then you're super comfortable saying "true probability" yourself. Why is "correct probability" weasel words but "true probability" isn't?

There's no ludic fallacy going on here, because there's no real life scenario here. There's only the thought experiment, and nobody's trying to apply it to real life.

[Skepdick]

I don't understand how you can say "correct probability" is weasel words, and then you're super comfortable saying "true probability" yourself. Why is "correct probability" weasel words but "true probability" isn't?

Because it's either true that I chose the 100/100 box; or it's true that I chose the 100/1 box.

And so straight-forward counter-factual reasoning follows.

And so IF I happen to be in box 100/1 then 0.666 estimation that the next note is a 100 is wrong.

There's no ludic fallacy going on here, because there's no real life scenario here. There's only the thought experiment, and nobody's trying to apply it to real life.

That's what I fucking said! There's no idea on the table.

[Flannel Jesus]

So, here's the whole thing then:

Some people analytically conclude that the probability is 50%
I can analytically calculate the 2/3 probability.
Your code that you wrote experimentally verifies that the probability hovers around 2/3 when tried many times, in the long run.

So now, when Bob and I refer to 2/3 as "correct" and 50% as incorrect, it's weasel words and fallacies? It's somehow "tautological"?

So there's no way to be incorrect about the probability of something? Anybody could say any probability, and they're all equally correct, and there's no way analytically or experimentally to show they're incorrect?

Forget about the weighted coin, that was probably a bad example. I was trying to illustrate something but I don't think that example illustrated what I intended it to. Just keep it on this one, the boxes with the money. It's your position that there's no "correct probability" and that everybody's answer is equally valid? Is that it?

[Skepdick]

So, here's the whole thing then:

Some people analytically conclude that the probability is 50%
I can analytically calculate the 2/3 probability.
Your code that you wrote experimentally verifies that the probability hovers around 2/3 when tried many times, in the long run.

My code makes a fuckton of assumptions which may or may not be correct!
It assumes that you'll play the game a million times.
And so the implication of the calculation is that you would guess the true probability correctly 66% of the time IF you played over and over.

But if I set n=1 in the code and re-run the algorithm the answer is 0! Either you guessed right or wrong. There's no second try!

So now, when Bob and I refer to 2/3 as "correct" and 50% as incorrect, it's weasel words and fallacies? It's somehow "tautological"?

Yes. It's a tautology! You are only speaking in the internal language of the theory!

IF you happen to find yourself in the 100/1 box both the 50% and the 66% estimates are over-estimates. They are both wrong.

You can account for their "disagreement" as one using n=1 and the other using n=∞

So there's no way to be incorrect about the probability of something? Anybody could say any probability, and they're all equally correct, and there's no way analytically or experimentally to show they're incorrect?

You can say whatever the hell you want. Without some over-arching optimisation strategy or external feedback your notions of "correct" and "incorrect" aren't... (how shall we say)... interesting.

Forget about the weighted coin, that was probably a bad example. I was trying to illustrate something but I don't think that example illustrated what I intended it to. Just keep it on this one, the boxes with the money. It's your position that there's no "correct probability" and that everybody's answer is equally valid? Is that it?

Precisely that.

You are confusing time-averages (single run of the algorithm) with ensamble averages (a million runs of the algorithm).