## A fun little probability puzzle for you.

What is the basis for reason? And mathematics?

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Flannel Jesus
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### Re: A fun little probability puzzle for you.

Age wrote: Tue Jul 26, 2022 12:40 am Okay.

We have one box, and within it is one bill, which could either be a \$100 bill or a \$1 bill. So, how exactly could the probability be anything different to 50% that the next bill in that box is a \$100 bill?
First, I want to acknowledge that there is an intuitive approach to the problem that you can take that will give you the 50%. It's intuitive, it makes sense, it's not some otherworldly explanation that only someone irrational could come up with.

But the point of these sorts of thought experiments, largely, is to challenge our intuitions. Just because a particular approach is intuitive and makes sense doesn't make it correct.

I'll requote the original problem here:
Flannel Jesus wrote: Sat Jul 16, 2022 7:36 am I present to you 4 USD bills, 1 \$1 bill and 3 \$100 bills. I then tell you I'm going to put 2 of these bills in one box, and the other 2 in another box - I shut a curtain and do so out of your sight. I then present you with the two boxes.

So, in front of you now are 2 boxes, both apparently identical from the outside, but one has a \$1 and a \$100 in it, and the other has a \$100 and another \$100 in it. You don't know which one is which.

Now I say, you may choose a box, so you do so - I put the other box away. I now say, reach inside and grab one of the bills inside the box. You do so, and you find that you've selected a \$100.

What is the probability that the other bill remaining in the box you selected is also a \$100?
One way to reword the final question is, "What is the probability you selected the box with 2x100", because the only way that the other bill remaining in the box you selected is also \$100 is if, and only if, you selected the box with 2x100.

And the intuitive idea is, there's 3 ways to select a \$100 first, and 2 of those ways are if you selected the 100+100 box. So once you know you did select a \$100 first, you have a 2/3 chance of having the 100+100 box in front of you, and therefore a 2/3 chance of selecting another 100 next.

----

So what we have now are two intuitive explanations, one that gives 50%, one that gives 2/3. If that was all I had, was two competing intuitive explanations, I wouldn't know which one to choose.

But luckily, I have more than that to guide me.

So what do you think at this point?
Age
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### Re: A fun little probability puzzle for you.

Flannel Jesus wrote: Tue Jul 26, 2022 5:50 am
Age wrote: Tue Jul 26, 2022 12:40 am Okay.

We have one box, and within it is one bill, which could either be a \$100 bill or a \$1 bill. So, how exactly could the probability be anything different to 50% that the next bill in that box is a \$100 bill?
First, I want to acknowledge that there is an intuitive approach to the problem that you can take that will give you the 50%. It's intuitive, it makes sense, it's not some otherworldly explanation that only someone irrational could come up with.

But the point of these sorts of thought experiments, largely, is to challenge our intuitions. Just because a particular approach is intuitive and makes sense doesn't make it correct.
First, I want to acknowledge that there can be approaches to the thought experiment/question raised here that you could take that will give you the 66.6%.
It is an approach, it makes sense, well to you anyway, and it is not some otherworldly explanation that only someone irrational could come up with.

But the point of these sorts of thought experiments, largely, is to challenge the way you think about things. Just because a particular approach is seen as correct, and makes sense does not make it correct

See, sometimes, exactly like i have done here, and am still doing here, I look at the other approach/s, like the ones you are using here, then I think further about them, and then I come to a realisation and just express the conclusion I arrived at. Sometimes that realisation and conclusion is that the intuitve answer was the actual correct one from the start, and sometimes I realise and conclude that the intuitive one was not the correct one at all.

But what I do is not use the approach that you are using and doing here. See I do not come to a realisation nor conclusion and then believe them to be true, and then just argue nor fight for that already believed position.
Flannel Jesus wrote: Tue Jul 26, 2022 5:50 am I'll requote the original problem here:
Okay. But I have already done the thought experiment here and have already arrived at a conclusion. I am just expressing my conclusion, which is what you asked for in your opening post.

I also know what you are trying to show and prove. But because I have already looked at and considered that, or in other words included some of your approaches in my thinking and in your thought experiment, and this is how and why I have the conclusion that I have here now.

See, with three or more boxes, and still only one \$1 bill, your approach does give you more than 50%. But as it your thought experiment stands, with only two boxes to begin with, then I am only seeing 50% probability.

But this view may change as we move along here.

Either you will have to show me what I am missing here, or I will have to learn how to show you what I think you are missing here.
Flannel Jesus wrote: Sat Jul 16, 2022 7:36 am I present to you 4 USD bills, 1 \$1 bill and 3 \$100 bills. I then tell you I'm going to put 2 of these bills in one box, and the other 2 in another box - I shut a curtain and do so out of your sight. I then present you with the two boxes.

So, in front of you now are 2 boxes, both apparently identical from the outside, but one has a \$1 and a \$100 in it, and the other has a \$100 and another \$100 in it. You don't know which one is which.

Now I say, you may choose a box, so you do so - I put the other box away. I now say, reach inside and grab one of the bills inside the box. You do so, and you find that you've selected a \$100.

What is the probability that the other bill remaining in the box you selected is also a \$100?
Flannel Jesus wrote: Tue Jul 26, 2022 5:50 am One way to reword the final question is, "What is the probability you selected the box with 2x100", because the only way that the other bill remaining in the box you selected is also \$100 is if, and only if, you selected the box with 2x100.
So, the probability and answer here would also be 50%.
Flannel Jesus wrote: Sat Jul 16, 2022 7:36 am And the intuitive idea is, you know you've selected one of the \$100s, so what you do is you compare the relative likelihoods of having selected a \$100 first if you had chosen the 100+1 box, compared to if you have chosen the 100+100 box.
If that is what you did, then so be it. But I never did this. Or, in other words, what you say what I do, is actually not what I would do at all. Is this understood, by you?

Now, what I actually do, and did, is just see that I have one box in front of me right now, and the probability that that box contains a \$1bill or a \$100 bill is 50% because no matter which box I chose and took out a \$100 bill out of, there will always either be a 1\$ bill or \$100 bill in the box.
Flannel Jesus wrote: Tue Jul 26, 2022 5:50 am
If you had selected the 100+1 box, there's a 50% chance that you would have selected the \$100. If you had selected the 2x100 box, there's a 100% chance you would have selected a \$100.

So, there's double the chances of seeing the result you did see, if you had selected the 2x100 box compared to the 100+1 box.
But this plays no part in the probability of what is left in the box, as far as I can see here yet.
Flannel Jesus wrote: Tue Jul 26, 2022 5:50 am There's 3 ways to select a \$100 first, and 2 of those ways are if you selected the 100+100 box.
I do not see that this matters, in relation to the actual question you asked, and which I answered.
Flannel Jesus wrote: Tue Jul 26, 2022 5:50 am So once you know you did select a \$100 first, you have a 2/3 chance of having the 100+100 box in front of you.
But there was only two boxes to choose from. So, the probability that you picked that box was 50%.

But, if that is what you want to believe is true, then so be it.

Flannel Jesus
Posts: 222
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### Re: A fun little probability puzzle for you.

Age wrote: Wed Jul 27, 2022 4:34 am Okay. But I have already done the thought experiment here and have already arrived at a conclusion. I am just expressing my conclusion, which is what you asked for in your opening post.
My entire post was a response to you asking this: "So, how exactly could the probability be anything different to 50%?" That question is not just you expressing your conclusion, that question is you asking me to explain mine. In this conversation, right now, you are not "just expressing my conclusion", you are decidedly doing more things than that.
Age wrote:
Flannel Jesus wrote: wrote: ↑Sat Jul 16, 2022 7:36 am
And the intuitive idea is, you know you've selected one of the \$100s, so what you do is you compare the relative likelihoods of having selected a \$100 first if you had chosen the 100+1 box, compared to if you have chosen the 100+100 box.
If that is what you did, then so be it. But I never did this. Or, in other words, what you say what I do, is actually not what I would do at all. Is this understood, by you?
I did not say you did this. You may have misunderstood something, I do not think you did this.

This is the second of two intuitive approaches. One intuitive approach gives you your result of 50%, this is the other one, to arrive at 2/3.
Yes, you explained your intuitive approach, and I explained that there is an alternative intuitive approach.

So we have two intuitive approaches - one you agree with, to arrive at 50%, and one you don't agree with, to arrive at 2/3. What we need is something outside of intuitive approaches, something more rigorous that might tell us which one is right.

So, for me personally, I have resources available to me that would help me do that.

But before we do that, it seems you think my intuitive approach, to produce 2/3, has some very non-intuitive steps:
Age wrote:
Flannel Jesus wrote:So, there's double the chances of seeing the result you did see, if you had selected the 2x100 box compared to the 100+1 box.
But this plays no part in the probability of what is left in the box, as far as I can see here yet.
I have an explanation for why this does play a part, and it is by way of analogy. Do you recall the analogy of 2 boxes with 50 balls each, one with 50 blue balls and one with one blue ball and 49 red balls?
Age
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### Re: A fun little probability puzzle for you.

Flannel Jesus wrote: Wed Jul 27, 2022 6:38 am
Age wrote: Wed Jul 27, 2022 4:34 am Okay. But I have already done the thought experiment here and have already arrived at a conclusion. I am just expressing my conclusion, which is what you asked for in your opening post.
My entire post was a response to you asking this: "So, how exactly could the probability be anything different to 50%?" That question is not just you expressing your conclusion, that question is you asking me to explain mine. In this conversation, right now, you are not "just expressing my conclusion", you are decidedly doing more things than that.
Age wrote:
Flannel Jesus wrote: wrote: ↑Sat Jul 16, 2022 7:36 am
And the intuitive idea is, you know you've selected one of the \$100s, so what you do is you compare the relative likelihoods of having selected a \$100 first if you had chosen the 100+1 box, compared to if you have chosen the 100+100 box.
If that is what you did, then so be it. But I never did this. Or, in other words, what you say what I do, is actually not what I would do at all. Is this understood, by you?
I did not say you did this. You may have misunderstood something, I do not think you did this.
When you write and use words like;
"you know you have selected ... so what you do is you compare". Who does the 'you' word here refer to, exactly?
Flannel Jesus wrote: Wed Jul 27, 2022 6:38 am This is the second of two intuitive approaches. One intuitive approach gives you your result of 50%, this is the other one, to arrive at 2/3.
Are all approaches intuitive approaches?

Flannel Jesus wrote: Wed Jul 27, 2022 6:38 am
Yes, you explained your intuitive approach, and I explained that there is an alternative intuitive approach.
But you appear to not yet recognize what the other approach is, which I was and have been using.
Flannel Jesus wrote: Wed Jul 27, 2022 6:38 am So we have two intuitive approaches - one you agree with, to arrive at 50%, and one you don't agree with, to arrive at 2/3.
Why do you call these approaches, intuitive approaches?
Flannel Jesus wrote: Wed Jul 27, 2022 6:38 am What we need is something outside of intuitive approaches, something more rigorous that might tell us which one is right.
Something outside of an intuitive approach is what I hope you will find is exactly what I have been doing.
Flannel Jesus wrote: Wed Jul 27, 2022 6:38 am So, for me personally, I have resources available to me that would help me do that.
For me, I also have resources available to me that did help me do something more rigorous, which did tell me which one is right.
Flannel Jesus wrote: Wed Jul 27, 2022 6:38 am But before we do that, it seems you think my intuitive approach, to produce 2/3, has some very non-intuitive steps:
Why do you keep using the intuitive word here?

What is this intuitiveness or intuition in relation to, exactly?
Flannel Jesus wrote: Wed Jul 27, 2022 6:38 am
Age wrote:
Flannel Jesus wrote:So, there's double the chances of seeing the result you did see, if you had selected the 2x100 box compared to the 100+1 box.
But this plays no part in the probability of what is left in the box, as far as I can see here yet.
I have an explanation for why this does play a part, and it is by way of analogy. Do you recall the analogy of 2 boxes with 50 balls each, one with 50 blue balls and one with one blue ball and 49 red balls?
Yes, and I also know why this does not work in relation to the other scenario in your opening post. But I can see very clearly why you think and presume that it does.
Flannel Jesus
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### Re: A fun little probability puzzle for you.

Please, if you have extra resources beyond your intuitive explanation that allowed you to prove it is right to yourself, present them.
Age
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### Re: A fun little probability puzzle for you.

Flannel Jesus wrote: Wed Jul 27, 2022 12:06 pm Please, if you have extra resources beyond your intuitive explanation that allowed you to prove it is right to yourself, present them.
The Fact that there are only two things to choose from proves it is right. Nothing else is needed
Flannel Jesus
Posts: 222
Joined: Mon Mar 28, 2022 7:09 pm

### Re: A fun little probability puzzle for you.

Age wrote: Thu Jul 28, 2022 2:34 am The Fact that there are only two things to choose from proves it is right. Nothing else is needed
I don't think you've given me enough compelling reasons to change my mind. You've given me a sense of your intuitive argument, and not more, but my position does have more. Here's what I have to support 2/3:

An alternative intuitive argument.
A bayesian calculation.
An experiment in code.
A consistency between all three.
A consistency in how I handle other similar situations (for example, the 2 bags with 50 balls problem).

If all you have to offer is "The Fact that there are only two things to choose from proves it is right. Nothing else is needed", then of course you're free to find that compelling, but I'm not going to. I have far more things in support of my position that you'd need to break down before I would be compelled to change my mind.
Age
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### Re: A fun little probability puzzle for you.

Flannel Jesus wrote: Thu Jul 28, 2022 7:13 am
Age wrote: Thu Jul 28, 2022 2:34 am The Fact that there are only two things to choose from proves it is right. Nothing else is needed
I don't think you've given me enough compelling reasons to change my mind
1. You do not have a mind.

2. If you did, then I would not even be wanting you to change it.

3. I am not here trying to convince you of absolutely anything nor to change absolutely any view you may have here either

You wrote;
"What is the probability that the other bill remaining in the box you selected is also a \$100?"

'Is it 50%?'

If you can not see that if, when making a choice, and there are only two things to you choose between, or from, and that the probability that you get one of those two different things is 50%, then okay. Or, if you do not find this Fact a compelling reason, then also okay.
Flannel Jesus wrote: Thu Jul 28, 2022 7:13 am You've given me a sense of your intuitive argument, and not more, but my position does have more. Here's what I have to support 2/3:
I asked you before to clarify why you use the intuitve word. Did you answer that question?
Flannel Jesus wrote: Thu Jul 28, 2022 7:13 am An alternative intuitive argument.
A bayesian calculation.
An experiment in code.
A consistency between all three.
A consistency in how I handle other similar situations (for example, the 2 bags with 50 balls problem).
Is there a probability that any of these are wrong in relation the actual question you asked for clarification?

If yes, then what is that probability?

But if no, then are you absolutely sure?

Also, what is the probability in the 50 ball so-called problem?

Flannel Jesus wrote: Thu Jul 28, 2022 7:13 am If all you have to offer is "The Fact that there are only two things to choose from proves it is right. Nothing else is needed", then of course you're free to find that compelling, but I'm not going to.
As I said above, then okay.

And, as I have also said, you are absolutely free to find absolutely anything compelling also.
Flannel Jesus wrote: Thu Jul 28, 2022 7:13 amI have far more things in support of my position that you'd need to break down before I would be compelled to change my mind.
I have broken them down, and to me the Fact that there are only two different things that you are choosing from always leads me back to you have a 50% chance of choosing one of them.

A Fact is irrefutable.

Therefore, I do not need anymore support.

To me, pure simplicity works just fine here.
Flannel Jesus
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### Re: A fun little probability puzzle for you.

Age wrote: Thu Jul 28, 2022 8:01 am I asked you before to clarify why you use the intuitve word. Did you answer that question?
I mean you've given me a casual, non-rigorous explanation in layman's terms of how your reasoning goes.
Age wrote: Thu Jul 28, 2022 8:01 am Is there a probability that any of these are wrong in relation the actual question you asked for clarification?

If yes, then what is that probability?
Of course I could be wrong. I don't know how to answer "what is the probability I'm wrong" though.
Age wrote: Thu Jul 28, 2022 8:01 am I have broken them down, and to me the Fact that there are only two different things that you are choosing from always leads me back to you have a 50% chance of choosing one of them.

A Fact is irrefutable.
So when there are two options, it's always 50/50 no matter what. So if I ask "what's the probability you have AIDS?" it's 50/50, since you either have AIDS or you do not. Is that correct?

If you're not interested in any evidence or arguments that you may be wrong, then... well, I thought you said you were here to learn, so that's a bit surprising. What do you hope to get out of this conversation? You're not willing to give me any better evidence to change my mind, and you're not willing to accept anything that might change yours, so what else is there to talk about?
Age
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### Re: A fun little probability puzzle for you.

Flannel Jesus wrote: Thu Jul 28, 2022 9:11 am
Age wrote: Thu Jul 28, 2022 8:01 am I asked you before to clarify why you use the intuitve word. Did you answer that question?
I mean you've given me a casual, non-rigorous explanation in layman's terms of how your reasoning goes.
How much more formal, rigorous explanation, in any term, would you like a Fact to be given to you?

Obviously, if 'it' is a Fact, then 'it' is irrefutable. For example, if there are only two things to choose from, then the probability of one of them being chosen is 50%.

If any one wants to dispute this Fact, then please go ahead and explain how and why.
Flannel Jesus wrote: Thu Jul 28, 2022 9:11 am
Age wrote: Thu Jul 28, 2022 8:01 am Is there a probability that any of these are wrong in relation the actual question you asked for clarification?

If yes, then what is that probability?
Of course I could be wrong. I don't know how to answer "what is the probability I'm wrong" though.
But I never asked, "What is the probability that you are wrong?", absolutely anywhere.

What I actually asked can be seen, in what you quoted me as saying above. And what can be clearly seen is that I never asked you any such thing as you have imagined and assumed here.
Flannel Jesus wrote: Thu Jul 28, 2022 9:11 am
Age wrote: Thu Jul 28, 2022 8:01 am I have broken them down, and to me the Fact that there are only two different things that you are choosing from always leads me back to you have a 50% chance of choosing one of them.

A Fact is irrefutable.
So when there are two options, it's always 50/50 no matter what.
This will always depend on what has occurred previously, leading up to this point.
Flannel Jesus wrote: Thu Jul 28, 2022 9:11 am So if I ask "what's the probability you have AIDS?" it's 50/50, since you either have AIDS or you do not. Is that correct?
As I just said and pointed out, this will depend on what has occurred previously, leading up to this point.
Flannel Jesus wrote: Thu Jul 28, 2022 9:11 am If you're not interested in any evidence or arguments that you may be wrong, then... well, I thought you said you were here to learn, so that's a bit surprising.
What exactly am I here 'to learn'?

Also, I have already looked at the so-called evidence or arguments that you have put forward, and it is because of them that my answer of 50% came to be.

Before that I was just asking you if the answer is 50%.

See, for a while there I thought the answer was 66.6%, but then from what you have provided so far, this made me look further into this, from which I then saw and concluded the 50% answer.
Flannel Jesus wrote: Thu Jul 28, 2022 9:11 am What do you hope to get out of this conversation?
I was going to ask you the exact same question. Especially considering that it was you who started this conversation.

But what I hope to get out of this conversation is the actual and irrefutable True answer, to the actual question that you posed here.
Flannel Jesus wrote: Thu Jul 28, 2022 9:11 am You're not willing to give me any better evidence to change my mind, and you're not willing to accept anything that might change yours, so what else is there to talk about?
1. There is no your, nor my, mind.

2. I am not concerned one iota whether you change from what you believe is absolutely true here to something else.

3. I have already looked at and considered all the things that you have put forward, but that none of them actually proves what you believe and claim is true does not seem to concern you. You just believe that 66.6% is the one and only absolute true, right, and correct answer, right?

Oh, and by the way, if you get to prove that 66.6% is the only one absolute true, right, and correct answer, then what happens?
Flannel Jesus
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### Re: A fun little probability puzzle for you.

Okay, you've seen all the evidence and you think the answer is 50%. If you are still committed to that conclusion, despite Bayes theorem showing it's 66%, despite the experimental code provided by skepdick showing it's 66%, then there's nothing more I have to show you.

If you don't have anything more to show me either, then we're at the end.
Age
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### Re: A fun little probability puzzle for you.

Flannel Jesus wrote: Thu Jul 28, 2022 11:02 am Okay, you've seen all the evidence and you think the answer is 50%. If you are still committed to that conclusion, despite Bayes theorem showing it's 66%, despite the experimental code provided by skepdick showing it's 66%, then there's nothing more I have to show you.

If you don't have anything more to show me either, then we're at the end.
I have more to show you, but do you really want to see it?
Flannel Jesus
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### Re: A fun little probability puzzle for you.

Of course, if you have good stuff that is legitimately challenging, let's see it.
Age
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### Re: A fun little probability puzzle for you.

Flannel Jesus wrote: Thu Aug 04, 2022 2:12 pm Of course, if you have good stuff that is legitimately challenging, let's see it.
We have here two boxes in front of you. One is on your left and one is on your right. One of the boxes has one \$100 bill and \$1 bill in it, while the other box has a two \$100 bills it.

Which box do you pick, the one on your left or the one on your right?

(Oh, and by the way, if this is so-called 'good stuff' and/or
'legitimately challenging' to you, or not, is very relative. So, we will have to wait and see if I have the 'good stuff' that is 'legitimately challenging', or not.)
Flannel Jesus
Posts: 222
Joined: Mon Mar 28, 2022 7:09 pm

### Re: A fun little probability puzzle for you.

I'll pick the one on my left.