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.

I have already explained why the answer to me is 50%.