A fun little probability puzzle for you.

[bobmax]

Well, my mans is on fire, you got it right. I must have misunderstood your reasoning process

My mind is never really clear. It is not easy to determine what I really know.

The difficulty with probabilities, in which I continually stumble, I think mainly depends on not knowing what I really know.

[Skepdick]

i feel Monty Hall rolling over in his grave.

No, this problem is not equivalent.

In the Monty Hall scenario the host has perfect information.

Because the host has information - you gain information from their act. They'll never open the door where the car is!

[Skepdick]

I think the $100 does give us (incomplete, imperfect, probabilistic) information about which box we initially selected

It doesn't. That is the illusion of control. The odds of drawing a $100 the first time are always 3:1 - 0.75 probability.

The thing to realise with your problem is that the boxes and the choosing are just fluff. Irrelevant details.
The problem is equivalent to having one box with 4x notes.

If you take a note out and it's $100, what remains is 2x $100 + 1x $1 notes.
Odds of drawing a 100 next are 2:1 = 0.666... probability.

If you take a note out and it's $1, what remains is 3x $100 notes.
Odds of drawing a 100 next are 3:0 = 1 probability. An absolute certainty.

[Flannel Jesus]

I think the $100 does give us (incomplete, imperfect, probabilistic) information about which box we initially selected

It doesn't. That is the illusion of control. The odds of drawing a $100 the first time are always 3:1 - 0.75 probability.

The thing to realise with your problem is that the boxes and the choosing are just fluff. Irrelevant details.
The problem is equivalent to having one box with 4x notes.

If you take a note out and it's $100, what remains is 2x $100 + 1x $1 notes.
Odds of drawing a 100 next are 2:1 = 0.666... probability.

If you take a note out and it's $1, what remains is 3x $100 notes.
Odds of drawing a 100 next are 3:0 = 1 probability. An absolute certainty.

So then what do you think of the rest of the comment that you quoted?

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]

So then what do you think of the rest of the comment that you quoted?

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?

You are asking a very different question from the first example.

Trying to guess the denomination of the next note, or the color of the next ball is not the same thing as trying to guess which bag/box you have chosen.
Those are different probabilities.

Picking up a blue ball on first attempt will boost your confidence that you've chosen from the all-blue balls bag.

Picking up the very next ball will either absolutely affirm or absolutely destroy your confidence.

[Flannel Jesus]

You are asking a very different question from the first example.

Trying to guess the denomination of the next note, or the color of the next ball is not the same thing as trying to guess which bag/box you have chosen.
Those are different probabilities.

No no no, they are literally synonymous questions. The wording of the question might be different, as it focuses on a different element, but they have entirely parallel answers, the anwers are paired with each other 100% of the time.

Will the next bill be a 100$ bill?
Did you choose the box with 2 100$ bills?

Every time, literally every time, the answer to the first question is "yes", the answer to the second question is also "yes".
And every time the answer to the first question is "no", the answer to the second question is also "no".

So the wording of the questions is different, one focuses on the box and the other focuses on the next bill in the box, but the questions are fully parallel and the probabilities are fully synonymous.

Why do you think they're different? Can you describe a scenario where the answer to one question gives you a different answer to the other question?

Picking up a blue ball on first attempt will boost your confidence that you've chosen from the all-blue balls bag.

Yes, I'm glad you see this. In the same vein, picking a $100 bill should boost your confidence that you've chosen from the 2x100 box.

[Age]

The actual extraction changes nothing of this situation.

Instead, one is mistakenly led to believe that this extraction has an influence.

What extraction? You mean the fact that you selected the $100 bill changes nothing? Of course it does, it changes everything. The alternative is that the first bill you selected was a $1, in which case you'd be guaranteed that the remaining bill is $100

what I was trying to say is that the probability of taking a $ 100 bill out of the box with two $ 100 is twice that of taking it out of the other box.

Oh my mistake, you saying that you "don't have any extra box related knowledge" made me think you were saying 50/50. This happens to be correct, it's a 1/3 chance to select a 1 and a 2/3 chance to select a 100

Are you 100% absolutely sure that this is correct?

[Flannel Jesus]

The actual extraction changes nothing of this situation.

Instead, one is mistakenly led to believe that this extraction has an influence.

What extraction? You mean the fact that you selected the $100 bill changes nothing? Of course it does, it changes everything. The alternative is that the first bill you selected was a $1, in which case you'd be guaranteed that the remaining bill is $100

what I was trying to say is that the probability of taking a $ 100 bill out of the box with two $ 100 is twice that of taking it out of the other box.

Oh my mistake, you saying that you "don't have any extra box related knowledge" made me think you were saying 50/50. This happens to be correct, it's a 1/3 chance to select a 1 and a 2/3 chance to select a 100

Are you 100% absolutely sure that this is correct?

I'm not 100% sure of many things -- maybe not anything. I'm reasonably certain of this, probably more than 99% certain. I have multiple avenues of support for it - empirical support (thanks skepdick for the code), bayesian calculations, I also know that it's the standard accepted answer. So I'm reasonably certain, approaching 100% but never there.

[Skepdick]

No no no, they are literally synonymous questions. The wording of the question might be different, as it focuses on a different element, but they have entirely parallel answers, the anwers are paired with each other 100% of the time.

No, they aren't!

Asking "Which box DID I choose?" is asking a question about the past.
Asking "Which note WILL I pull out next?" is asking a question about the future.

[Age]

What extraction? You mean the fact that you selected the $100 bill changes nothing? Of course it does, it changes everything. The alternative is that the first bill you selected was a $1, in which case you'd be guaranteed that the remaining bill is $100


Oh my mistake, you saying that you "don't have any extra box related knowledge" made me think you were saying 50/50. This happens to be correct, it's a 1/3 chance to select a 1 and a 2/3 chance to select a 100

Are you 100% absolutely sure that this is correct?

I'm not 100% sure of many things -- maybe not anything. I'm reasonably certain of this, probably more than 99% certain. I have multiple avenues of support for it - empirical support (thanks skepdick for the code), bayesian calculations, I also know that it's the standard accepted answer. So I'm reasonably certain, approaching 100% but never there.

That is good.

I am just MISSING what you are FINDING and SEEING.

[Flannel Jesus]

No no no, they are literally synonymous questions. The wording of the question might be different, as it focuses on a different element, but they have entirely parallel answers, the anwers are paired with each other 100% of the time.

No, they aren't!

Asking "Which box DID I choose?" is asking a question about the past.
Asking "Which note WILL I pull out next?" is asking a question about the future.

Okay, so can you describe to me a scenario where the answer to one is not the same as the answer to the other?

[Flannel Jesus]

Are you 100% absolutely sure that this is correct?

I'm not 100% sure of many things -- maybe not anything. I'm reasonably certain of this, probably more than 99% certain. I have multiple avenues of support for it - empirical support (thanks skepdick for the code), bayesian calculations, I also know that it's the standard accepted answer. So I'm reasonably certain, approaching 100% but never there.

That is good.

I am just MISSING what you are FINDING and SEEING.

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

Empirical support via Slepdick here: viewtopic.php?p=583875#p583875
I translated it to javascript to be runnable in the browser here: https://jsfiddle.net/hbxnj850/1/

Standard answer shown here: https://en.wikipedia.org/wiki/Bertrand%27s_box_paradox
" probability is actually 2/3"

[Age]

I'm not 100% sure of many things -- maybe not anything. I'm reasonably certain of this, probably more than 99% certain. I have multiple avenues of support for it - empirical support (thanks skepdick for the code), bayesian calculations, I also know that it's the standard accepted answer. So I'm reasonably certain, approaching 100% but never there.

That is good.

I am just MISSING what you are FINDING and SEEING.

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

Empirical support via Slepdick here: viewtopic.php?p=583875#p583875
I translated it to javascript to be runnable in the browser here: https://jsfiddle.net/hbxnj850/1/

Standard answer shown here: https://en.wikipedia.org/wiki/Bertrand%27s_box_paradox
" probability is actually 2/3"

Instead of making complex, or confusing, what is essentially truly simple and easy I prefer to just keep things really simple and easy.

If I present you with a box that has either a $100 dollar bill or a $1 dollar bill in it, then what is the probability that the bill in the box is the $100 bill?

That is how I FIND and SEE what your question is essentially asking for in your opening post. So, what is 'it', EXACTLY, that I am MISSING here?

[Flannel Jesus]

I don't know what you're talking about.

[Skepdick]

Okay, so can you describe to me a scenario where the answer to one is not the same as the answer to the other?

Yes. Trivially.

Having picked a random box what's the probability it's the box containing only blue balls? 0.5
Having picked a random box what's the probability the first ball I take out is blue? 0.51

0.5 != 0.51