A fun little probability puzzle for you.

[Age]

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

Okay, but that might just be because what I SEE is just too SIMPLE for some?

If absolutely ANY one does NOT know what I am talking about, then they are completely FREE to ASK me absolutely ANY CLARIFYING questions AT ALL.

Anyway, until then, I suggest you LOOKING AT the ACTUAL question you posed and asked in your opening post and noticing that there is only ONE box, and that it could ONLY be holding within it either a $100 bill or a $1 bill.

Also, take notice that in your example there are NOT four DIFFERENT denominated bills, AND that there are only TWO boxes and NOT THREE boxes like in "bertrand's box" example. Also, the so-called 'empirical support' is based on calculations, which were made upon what "skepdick" programmed them to be of, which may have just been made on the SAME Wrong ASSUMPTIONS that you are making here.

Oh, and by the way, you also SAID and WROTE;

Don't look up the answer or read further, try to solve it yourself first.

So, I just did that.

[Flannel Jesus]

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 with 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

That's not the two questions. You've rephrased it to create a strawman.

[Skepdick]

That's not the two questions. You've rephrased it to create a strawman.

So what are the questions?

Formulate them in Mathematics.

[Flannel Jesus]

That's not the two questions. You've rephrased it to create a strawman.

So what are the questions?

Formulate them in Mathematics.

You said the two questions are

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

You phrase both of the questions from the point of view of someone who has not selected the first ball (or bill) already.

The question is NOT from the point of view of the person who has not yet selected a ball (or bill). The question is from the perspective of someone who HAS selected the first ball (or bill) and the first ball was blue (or the bill was $100).

So it's not
Having picked a random box what's the probability it's the box with blue balls?

It's
Having picked a random box and a random ball, and the ball was blue, what's the probability it's the box with blue balls?

It's not
Having picked a random box what's the probability the first ball I take out is blue?

It's
Having picked a random box and a random ball, and the ball was blue, what's the probability the SECOND ball I take out is blue?

[Skepdick]

Having picked a random box and a random ball, and the ball was blue, what's the probability it's the box with blue balls?

Having picked a random box and a random ball, and the ball was blue, what's the probability the SECOND ball I take out is blue?

≈0.980567 to both

Code: Select all

from random import sample
boxes = [ [ ["blue"] * 50 ] ,[ ["white"] * 49 + [ "blue"] ] ]

box_was_all_blue = 0
box_was_mostly_white = 0

# Perform the experiment a million times
for _ in range(1000000):
    box = sample(boxes, 1)[0][0] # Grab a random box
    ball = sample(box, 2)  # Grab random two balls 
    if ball[0] == "blue": # If the first ball was blue
        if ball[1] == "blue": # ...and the 2nd ball was blue
            box_was_all_blue +=1
        else: #... and the 2nd ball was white
            box_was_mostly_white +=1
    else:
        pass

probability = float(box_was_all_blue) / (box_was_all_blue + box_was_mostly_white)
print(probability)

[Flannel Jesus]

So the question still stands for you 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?

[Skepdick]

So the question still stands for you 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?

Which scenario?

The unconditional scenario, or the conditional scenario?

[Flannel Jesus]

Nevermind, it seems that the line of conversation was lost haha. I'll move past it lmao.

[Skepdick]

Nevermind, it seems that the line of conversation was lost haha. I'll move past it lmao.

Like. You know the answer here.

It's still Bayes.

Unconditional scenario means your prior is 0.5 ( https://en.wikipedia.org/wiki/Principle ... um_entropy )
Conditional scenario means you prior is 0.980567

[Flannel Jesus]

I'm not asking you for a number right now. You've lost the thread of conversation, but that's okay. I'm just gonna forget about it.

[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.

Nevermind, I'm getting bored with my day so i'm reopening it. This is what opened the conversation in the first place.

You said, pulling out the $100 gives us NO information about what box we initially selected.

BUT, you also experimentally proved that, once we've pulled the first $100, we have a 0.666.. probability that the next one will also be a $100.

So what's weird is, you're proving the point that you're disagreeing with. "I selected a $100, and the next bill will also be a $100" is 100% entirely synonymous with "I selected a $100, and I initially selected the box with 2x $100". Because the next bill CANNOT be a $100 unless you initially selected the box that has 2x $100. That's the only way for the next bill to be $100.

So, when you say "there's a 66% chance that the next bill is $100 after you select the first bill and it's $100", you MUST be saying "there's a 66% chance that you selected the box with 2x $100 after you select the first bill and it's $100"

Right? How could you possibly NOT be saying that?

[Skepdick]

BUT, you also experimentally proved that, once we've pulled the first $100, we have a 0.666.. probability that the next one will also be a $100.

Yes. And I also have 0.333... probability that the next one will be $1. We are in the land of possibilities (epistemology), not actualities (ontology).

And so the question "Am I in the box with two $100 notes?" (which is a yes/no question thus requiring 1 bit of information) could be interpreted as a question about epistemology; or a question about ontology. In your interpretation/view does probability of 0.666... amont to 1 bit information for answering "Am I in the box with two $100 notes?"

The confusion is simply in the fact that the term "information" is overloaded.

Information is that which resolves uncertainty.
Which uncertainty? There are a bunch at play.

[Flannel Jesus]

And so the question "Am I in the box with two $100 notes?" (which is a yes/no question ergo requiring 1 bit of information) cannot be answered.

But you gave an answer to "Will the next bill be $100?" and the answer was "with 66% probability, yes".

"Will the next bill be $100?" and "Did I choose the box with 2x $100?" are the same question. Every time one is true, the other is true. Every time one is false, the other is false.

So if the answer to one is "with 66% probability, yes", then the answer to the other must also be "with 66% probability, yes".

[Skepdick]

But you gave an answer to "Will the next bill be $100?" and the answer was "with 66% probability, yes".

Precisely. It's a confidence interval. Not a correct answer.

"Will the next bill be $100?" and "Did I choose the box with 2x $100?" are the same question.

That's only coincidental, not incidental.

In both scenarios drawing two balls is sufficient to resolve any uncertainty about which box I am in.

The Bayesian update resulting from the 2nd observation propagates through the system allowing me to determine precisely which box I am in.

Even if I initially predicted that I am in the other one.

[Flannel Jesus]

But you gave an answer to "Will the next bill be $100?" and the answer was "with 66% probability, yes".

Precisely. It's a confidence interval. Not a correct answer.

Yes, that's literally the point. What's the precent likelihood? What's your confidence? I think you've constructed a strawman of the scenario in your head, or drastically misinterpreted something that was said.

I said

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

You said

It doesn't.

What I said is very much about a confidence interval, isn't it? It gives us information, but that information is incomplete, imperfect, probabilistic. Before we selected the $100, there's a 50/50 chance of which box we selected. After we selected the $100, we're now at 66.666..%, by your own empirical observations.

So, our confidence interval has changed, we've gained information about what box we've likely selected.