I’m not sure if I need to clarify this, but just in case: you don’t get to change boxes. You’re not selecting a bill from all of the available boxes. You’ve chosen a box, and you’ve chosen the first bill out of that box. There’s only 1 bill left in the box you’ve chosen, so you can’t just pick from any of the boxes left.
This is your post on the 3 box scenario:
So you have chosen a box and selected a $100 from it.
All of the boxes had at least 1 $100 bill in them.
So you don’t have any more information about what is left in the box you selected than the start of the scenario.
All you know is there are 5 bills remaining, 2 $1 bills and 3 $100 bills.
So the bill remaining in your chosen box could be 1 of 3 $100 bills, or 1 of 2 $1 bills.
So the probability of the bill being $100 is 3 of 5 (60%) and 2 of 5 (40%) for $1 bill.
I’ll go into detail for the bayesian approach soon.
However, you said “you don’t have any more information about what is left in the box you selected than the start of the scenario.” I don’t see how you can say that, given that your position seems to be the opposite of that.
At the start of the scenario, before you see the first 100, the odds that you select(ed) the 100+100 box is 1/3, right? But after you see the 100, you’re saying the odds that you selected the 100+100 box have gone up to 60%. Maybe we’re not using the word “information” in the same way as each other, but I call that information. If it changes the odds, it’s information.
And we both agree that seeing the 100 changes the odds.
Your question was the probability of the REMAINING BILL being $100, not which box of the 3 boxes.
The question is about the money, not about the box.
The questions, in my opinion, are synonymous, and I’ll tell you why.
Once you’ve seen the first 100, the question “is the other bill a $100?” always, every time and without exception, has the same answer as the question “did I choose the box with 100+100?”
Those two questions will always have the same answer. If it’s no for one, it’s no for the other. If it’s yes for one, it’s yes for the other.
Once you’ve seen the first $100, the odds for one question must be identical to the odds for the other question.
nm FJ
It’s kind of absurd.
If you choose a box.
You have a 50% chance or a 100% chance.
Choosing a $100 the first time does not change that statistic.
You still have a 50% chance or a 100% chance.
Saying something like you have a 75% chance is non sensical.
It’s either a 50% chance or 100% chance.
Medians don’t determine statistics.
Eh?
The calculation is about the BILLS, not about the boxes.
There are 3 boxes. The probability that you have selected the box with the 2 $100 bills is 33.333…% and that will not be known until the other bill is selected.
As it stands after picking the first bill, and it being $100 means that you have a 1 in 3 probability that your box is the box which started with the 2 $100 bills.
So you think the probability is 33.333…% ???
I told you what I think the probability is, and I did not say 33.3%.
Right, so you are not calculating the boxes!
There are 3 boxes and you selected 1 of them. You have 33.333…% of the boxes, of which ALL of them had a $100 bill in them. You don’t have any more information than the start, so your probability is 1 of 3 boxes, 33.333…%.
But we both know that the calculation is NOT about the boxes, it’s about the remaining BILLS, and there are 5 remaining bills, 3 of which are $100. So there is a 60% probability of the bill in your box being a $100 bill.
I will show you my calculations later, I’m not able to now. I use Bayes theorem to calculate the probability that I selected the 100+100 box, given that the first bill I pulled out is 100. It comes out to 50/50.
But if you just can’t wait, I encourage you to try out the Bayes theorem method yourself.
I don’t believe that either.
There’s a 50% chance that it’s a 50% chance or 100% chance.
That doesn’t mean at the end that there’s a 50% chance… or even a median like 75% chance.
It means what it says…
There’s a 50% chance that it’s 50% or 100%.
I understand 1/2 options is 50%.
So I know why you argue that way you do.
Think about this though.
There is no more likely place if you draw a hundred first.
Pulling the first $100 bill changes nothing, all of the boxes contained a $100 bill. If there were 5 $100 bills and 1 $1 bill and you pulled the $1 bill, then you would know that there are 5 bills remaining, and all of them are $100, so there is a 100% chance (5 of 5) of pulling a $100 bill. The boxes are not factored into that equation.
"Pulling the first $100 bill changes nothing, all of the boxes contained a $100 bill. "
But this is not what you say when they’re 2 bags with the blue and red balls, remember? You don’t say “both bags have a blue ball, so pulling out a blue ball changes nothing”. In that case, you correctly calculated the odds of having selected the bag full of blue balls after having selected a blue ball out of it.
There were 51 blue balls and 49 red balls.
The probability of pulling a blue ball was 51 of 100 for blue balls (51%) and 49 of 100 for red balls (49%).
That is EXACTLY how I am calculating the probability of pulling a $100 bill, 3 of 5 (60%)
This, motor, is the post you made where you used the correct logic.
Right, I stated that there was a 51% probability of picking a blue ball. That is because there were 51 out 100 balls that were blue.
I also stated that a DIFFERENT CALCULATION was 50 out of 51 (98%) of which bag you picked the ball from.
Notice that the first calculation is 51 of 100 and the second calculation is 50 out of 51???
In your 3 box scenario you are asking the equivalent of the 51 out of 100 calculation (3 out of 5 $100 bills remaining). You are not asking the calculation of a percent of 3 boxes.
You stated that probability of picking the FIRST ball was 51%. But nobody’s asking about the first ball.
It is a new scenario once the first $100 bill is picked because there are now only 5 bills remaining. It is equivalent to a brand new scenario that starts with 5 bills, 3 of which are $100.
Like I stated many times, EVERY TIME a bill is selected a new calculation is required. The NEW calculation is a new scenario, that which there is 5 bills, 3 of which are $100 and 2 of which are $1 bills.
There is no difference in what remains of the original 6 bills after pulling a bill, and a new scenario that STARTS with 5 bills. A NEW CALCULATION IS REQUIRED every single time the numbers change.