A fun little probability puzzle for you.

[Skepdick]

The idea that it's still 50/50 after seeing the $100 - empirically, it can be shown to hover around 66.66...%

Notice how you keep insisting on contextualising/framing/conditioning the scenario around "after seeing the $100 note".

Notice how you also keep insisting on contextualising/framing/conditioning the scenario around "after performing the experiment infinitely many times".

Only you aren't being as explicit about your 2nd condition as you are with your 1st.

Analytically, it can be shown to be 2/3, through various means but my preferred means is Bayes theorem.

Basically, you aren't being very analytical if you are blindly using a theorem you can't derrive from first principles.

You are just doing what Bayes told you to do, without any understanding of why you are doing it; or the conditions under which the theorem holds.

Good computer *pats on head* , here's a snack!

[Flannel Jesus]

Age, I also gave an analogy about why selecting something from the box can be said to give us probabilistic information about what is in the box.

Instead of two boxes with two bills each, imagine I had two bags with 50 coloured balls each. One of the bags has 50 blue balls. The other bag has 1 blue ball, and 49 red balls. I present you the two bags, you don't know which one is which, so you choose one bag at random. Then you stick your hand in and pick out one ball at random. The one ball you picked out is blue.

Does the fact that you selected a blue ball give you any possible information about what bag you selected? Are you more likely to have selected the bag full of blue balls, once you see that you've selected a blue ball? Or are you just as likely to have selected the bag with 49 red balls?

[Skepdick]

Age, I also gave an analogy about why selecting something from the box can be said to give us probabilistic information about what is in the box.

Instead of two boxes with two bills each, imagine I had two bags with 50 coloured balls each. One of the bags has 50 blue balls. The other bag has 1 blue ball, and 49 red balls. I present you the two bags, you don't know which one is which, so you choose one bag at random. Then you stick your hand in and pick out one ball at random. The one ball you picked out is blue.

Does the fact that you selected a blue ball give you any possible information about what bag you selected? Are you more likely to have selected the bag full of blue balls, once you see that you've selected a blue ball? Or are you just as likely to have selected the bag with 49 red balls?

Surprise is information and information is surprise.

To claim that something gave you information is the same as saying that you were surprised by it.
And to claim that something surprised you is the same as claiming that you didn't expect it to happen.

If you know that you have two boxes full of blue and white balls, it's a really pecular claim that taking any particular blue; or white ball should surprise you. It's expected and therefore it's not surprising at all.

There has to be something else at play here. You have to have some other (unstated) expectation/assumption at play here if any particular blue, or white ball surprises you!

The unstated assumption is entropy itself. Randomness. Fairness. Homogenous distributions. Uniformity.

The blind belief that all balls had equal opportunity of being chosen.

e.g the explicit denial of the fact that balls at the bottom of the bag are less likely to get chosen. The denial of inherent selection bias in the natural world.

It's an appeal to the Central limit theorem (and all of its axiomatic assumptions)

[Age]

Let's go into more detail on my Bayes theorem layout. This is what I initially said:

Bayes theorem is

P(A|B) = P(B|A) * P(A) / P(B)

A = I picked the box with 2 100$
B = I chose a $100

we want to find P(A|B), probability of A given B

P(B|A) = probability of B given A, which is 1
P(A) is 0.5
P(B) is 3/4

1 * 0.5 / (0.75)
2/3

P(B|A) = probability of B given A, which is 1
Do you agree that the probability that the bill I chose is a $100, IF I chose the box with 2x100, is 1? If not, why not?

Besides the Fact that you are, once again, complicating 'that' what is, essentially, PURE SIMPLE, you are here using words that just also conflate and/or just cause MORE CONFUSION.

When you say 'chose' did you mean 'will choose'?

P(A) is 0.5
Do you agree that the initial probability that I chose the box with 2x100 is 0.5? If not why not?

P(B) is 3/4
Do you agree that my first bill selection is 3/4 likely to be a $100? If not, why not?

Bayes theorem is formulated as
P(A|B) = P(B|A) * P(A) / P(B)
Do you agree with that? If not why not?

And when you plug in the above values into that formula, you get
1 * 0.5 / (0.75)
Do you agree with that? If not, why not?

Like most so-called 'paradoxes' you are just MISINTERPRETATING some words or ideas here.

In YOUR example there are ONLY TWO DIFFERENT denominated bills, although there are FOUR bills.

And this is WHAT and WHERE your MISUNDERSTANDING and CONFUSION here is arising from, EXACTLY.

[Flannel Jesus]

I don't think "chose" and "will choose" are meaningfully different in the part you've quoted, so if it makes you more comfortable to say "will choose", then feel free to do that

[Age]

The idea that it's still 50/50 after seeing the $100 - empirically, it can be shown to hover around 66.66...% likelihood of being the box with 100+100. Analytically, it can be shown to be 2/3, through various means but my preferred means is Bayes theorem.

There is ONLY 50/50 chance or probability.

In a thread with all of these avenues that have been laid out, you're just asserting that their false without looking at them.

LOL

I have LOOKED AT them. AND, KNOW WHERE you are Wrong. I can also SEE, EXACTLY, WHY you are SEEING things Wrong here.

In fact, I made this post to you here laying out bayes theorem and the code that runs the experiments:

viewtopic.php?p=583923#p583923

And your reply was "that's too complex for me".

Talk about BEING BLINDED by one's OWN BELIEFS and ASSUMPTIONS.

LOOK AT the ACTUAL WORDS that I USED and it is as CLEAR and as PLAIN as DAY that I NEVER said ANY such thing AT ALL. Which is PROVED absolutely and irrefutatbly True, Right, AND CORRECT by the ACTUAL WORDS written down there for ALL to LOOK AT and SEE.

So, what you SAY and CLAIM here could NOT be MORE False, Wrong, AND Incorrect.

If it's too complex for you, fine, no shame in that, but then you don't really stand on very solid ground when you say that the standard position can't be proven. It can't be proven because the proofs are too complex for you?

Talk about STUPIDITY in the EXTREME.

I suggest you READ from the Truly OPEN perspective and NOT from that VERY OBVIOUSLY CLOSED perspective you have here. That way you will NOT be AS Wrong as you CLEARLY ARE here.

[Flannel Jesus]

You're shouting a lot but you aren't saying much. I provided a lot of reasoning for why I think what I think. I'm very open to arguments about why that might be wrong, but you're not doing that, you're just shouting and insisting I'm wrong.

It doesn't matter how many times you insist it, it doesn't matter how heavily you lean on the caps lock key, if you don't engage with the arguments being made you aren't going to be very persuasive.

Relax, and persuade me

[Age]

I don't think "chose" and "will choose" are meaningfully different in the part you've quoted, so if it makes you more comfortable to say "will choose", then feel free to do that

If you REALLY can NOT SEE one 'happens' IN THE FUTURE, while the other 'happened' IN THE PAST, which will then absolutely EFFECT YOUR QUESTION, Do you agree that the probability that the bill I chose is a $100, ..., then you will also NOT SEE how they are VERY MEANINGFULLY DIFFERENT.

1. There is NO 'probability' AT ALL that the bill you 'chose' is a $100 IF you HAD ALREADY 'chosen' the $100 dollar bill. However,

2. There is A 'probability' that the bill you 'will choose' is a $100 IF you HAVE NOT YET ALREADY 'chosen' the $100 dollar bill.

[Flannel Jesus]

"if I chose the box with 2x100, the probability that the first bill I pull out is 1"

"if I will choose the box with 2x100, the probability that the first bill I pull out is 1"

As far as I can see, those two statements are both equally true for the same exact reasons. You're getting caught up on something very small here

[Age]

You're shouting a lot but you aren't saying much.

HERE IS ANOTHER COMPLETELY False AND Wrong ASSUMPTION of YOURS.

I provided a lot of reasoning for why I think what I think.

YET because you ARE CLOSED you are NOT SEEING WHY you are STILL Wrong.

I'm very open to arguments about why that might be wrong, but you're not doing that, you're just shouting and insisting I'm wrong.

YOUR ASSUMPTION here is STILL Wrong. And, I am ONLY 'insisting' you are Wrong, BECAUSE you ARE Wrong here.

If you EVER QUERIED me about my CLAIM that there are ONLY TWO ACTUALLY DIFFERENT bills in YOUR EXAMPLE, and it for this REASON WHY there will ONLY be a 50/50 chance or probablity, then you COULD SEE and UNDERSTAND WHY you are Wrong here.

It doesn't matter how many times you insist it, it doesn't matter how heavily you lean on the caps lock key, if you don't engage with the arguments being made you aren't going to be very persuasive.

LOL

'Projection' at its BEST.

Relax, and persuade me

But I can NOT 'persuade' you of absolutely ANY thing, which opposes what you currently BELIEVE is ABSOLUTELY TRUE.

[Flannel Jesus]

I've changed my mind about things countless times in my life time. Changed my mind drastically. I'm not immune to persuasion. I might be immune to your style of persuasion, which involves mostly shouting apparently, but not in general

[Age]

"if I chose the box with 2x100, the probability that the first bill I pull out is 1"

"if I will choose the box with 2x100, the probability that the first bill I pull out is 1"

As far as I can see, those two statements are both equally true for the same exact reasons. You're getting caught up on something very small here

I do NOT even KNOW WHY you are SAYING those two statements here.

They are ABSOLUTELY NOTHING I have EVER thought, let alone SAID.

You are NOWHERE NEAR what I have been SAYING, SHOWING, and POINTING OUT here.

[Flannel Jesus]

P(B|A) = probability of B given A, which is 1
Do you agree that the probability that the bill I chose is a $100, IF I chose the box with 2x100, is 1? If not, why not?

Besides the Fact that you are, once again, complicating 'that' what is, essentially, PURE SIMPLE, you are here using words that just also conflate and/or just cause MORE CONFUSION.

When you say 'chose' did you mean 'will choose'?

You had a problem with this wording. That's what we're talking about, right?

[Age]

I've changed my mind about things countless times in my life time. Changed my mind drastically. I'm not immune to persuasion. I might be immune to your style of persuasion, which involves mostly shouting apparently, but not in general

1. 'you' do NOT 'have a mind'.

2. CHANGING thoughts has NEVER been disputed.

3. Drastically CHANGING views is VERY COMMON. But this is usually because of what one previously BELIEVED was true.

4. NO one is immune to persuasion. You have just ONCE AGAIN completely MISSED and MISUNDERSTOOD what I have been SAYING and MEANING. But this is just because you have NEVER even attempted to GAIN ANY CLARITY AT ALL.

5. You are NOT immune to my so-called 'style of persuasion' AT ALL. I write in a way, SHOWING, and NOT necessarily PERSUADING.

6. I have NEVER 'shouted' ONCE. So, ONCE AGAIN, your continual ASSUMING here is STILL Wrong AND Incorrect.

This is just SO SIMPLE and EASY. The ONLY possibility in YOUR EXAMPLE is that there is either a $100 bill OR a $1 bill in that box. THEREFORE, the 'probability' IS and can ONLY BE a 50% CHANCE.

Which MEANS your 66% 'probability' or CHANCE is TOTALLY and UTTERLY False, Wrong, AND Incorrect.

How could ANY NOT SEE this NOW?

[Age]

P(B|A) = probability of B given A, which is 1
Do you agree that the probability that the bill I chose is a $100, IF I chose the box with 2x100, is 1? If not, why not?

Besides the Fact that you are, once again, complicating 'that' what is, essentially, PURE SIMPLE, you are here using words that just also conflate and/or just cause MORE CONFUSION.

When you say 'chose' did you mean 'will choose'?

You had a problem with this wording. That's what we're talking about, right?

1. I do NOT have 'a problem' with that wording, of YOURS. That wording of YOURS just CONFLATES and/or CONFUSES MORE.

2. In WHICH 'replies', "that is what we are talking about", are you referring to, EXACTLY?