I don't really think you know what I said. If you need clarification just ask [remainder of post removed by mod].

I noted towards the start of the thread that this had more to with defining the sample space. Calculating expected returns is a futile effort if we cannot agree on the underlying assumptions. The natural device for such a situation would be the Law of Parsimony, but I can't really say my approach makes fewer assumptions than Michael's. I do think, however, Occam's razor does cut additional assumptions about the distribution.

Statistics uses repeated observations of outcomes from a defined sample space, to make inference about the probability space associated with that sample space. — JeffJo

I just said that. That is exactly what I said.

Not even "one event," if that term were correct to use in your context. — JeffJo

I already posted the definition of an event from one of my books, which I will refer to over you. I will always go with my training over you.

The OP deals with a conceptual probability problem. There is no no observational data possible. "Statistics" does not belong in any discussion about it. — JeffJo

I also said that.


One thing I was taught in my first stats class was that the lexicon was not standardized. They told me to expect different people to use different terminology. I think perhaps you should take a page from that book. In the meantime, in my personal usage, I am sticking with the terminology in my books, no matter how much you protest.

You created a double standard. You can try to bury that in text, but that is what happened.

If you are using the principle of indifference then criticizing people for using the principle of indifference, that is hypocritical.

So you think you always have a 1.25 expected gain in every possible instance?

As far as I am concerned I already found the flaw. Take it or leave it, that is your choice.

Hey, if you feel lucky then switch, if you think you are close to the cap don't, feel this one out, but you are not going to be able to justly quantify a positive gain based on the information we have.

You are guessing. Then trying to model your guesswork.

It is also rational to want ice-cream on a hot day. You still don't know anything about the distribution.

Statistics is a data science and uses repeated random events to make inference about an unknown distribution. We don't have repeated random events, we have one event. Seems like a clear divide to me. You can't learn much of anything about an unknown distribution with just one event.

You don't know the distribution, you don't know the limits and you only get one chance to switch.

You only get one instance of the game. You don't get to open a 100+ envelopes.

Already commented on "strategy of switching". You are no longer talking about just probability anymore, since you can now sample the distribution you are now engaged in statistics, which is outside the scope of the OP

Actually only one case is true, while the other one does not exist. So they can't both be possible outcomes, not objectively. Remember the envelopes are already decided. You are modeling your assumption of what you think is possible. However, just because you can think of something that doesn't mean it is objectively a possible outcome.

I don't really care about things you can't know.

The 1.25X come from considering expected gains over both cases, the larger and smaller. However, when one case is true the other cannot be true, so since the chance event for the envelope has already been decided it makes no sense to consider expected gains in this fashion. They should to be considered separately.

Do you get the point or not?

Never said anything about both being actual at once. Never meant that at all. Clearly that can't happen so I am not sure how you got that. You have to consider each case on its own and not together. We had this same issue at the start of the thread.

That is what I just did. The envelopes cannot be in both cases at once, therefore it makes no sense to hedge your expections that both cases are possible. You need to either consider each case separately or just go off the fact you have two envelopes in front of you as the end results is the same.

And I am saying that doesn't really matter because it will always be amount A and amount B.

See that was easy.

I have two envelopes, one with amount A and one with amount B. I flip a fair coin to choose one. What is my chance of getting B?

I think we are safe, I doubt anything will blow up.

It absolutely can be ignored.

It is a subjective modeling based on expections when viewing Y. It confounds the objective with the subjective. The random event that determined the contents of the envelopes has already occurred and passed. That was the objective event. The selection is a separate event and it is subjective, which means these expectations are half-truths and everyone is just modeling their subjectivity. I am purposefully avoiding expected calculations for that reason. The envelopes are not in a possible state of two cases, they are in one case.

The filling of the envelopes and the selecting of the envelopes are two separate events.

Why don't you consider it from a categorical perceptive then. The OP never called for a solution based on expected gains.

Never said it was.

It has always been the same error, making assumptions based on Y. Seeing Y does not actually change the contents of the envelopes.

Yet you assume he is wrong?

I have a feeling though that Michael will still think that absent knowledge of the distribution, he can turn back to 50% as an assumption. — Srap Tasmaner

You have not really proven he can't. You yourself are making your own assumptions when considering expected gain over the two possible cases.

If a loaded coin flips H 9 out 10 times, without that knowledge, an uninformative 50/50 prior is completely justified.

Consider this: Say I have an unfair coin, on average it flips H 9 out of 10 times. You don't know this; however, all you see is a coin and without knowing it is unfair you give H a 50% chance. That is the difference between subjective and objective probability.

The only way for you to know that the coin flips H 9 out of 10 times is to flip the coin several times. Maybe I flip it for you a few times, say I get four heads in a row, and you are starting to doubt your 50/50 assumption. Then I flip it more and get two more heads, now you no longer believe it is 50/50.

That's Bayesian inference in a nutshell.

The amount in the envelopes is already set, just like the coin was already flipped, the uncertainty we have been considering is subjective in nature.

There is no reason you can't look at the OP and consider it in a categorical case.

Your reasoning would seem to suggest that there's a 50% chance of a coin flip landing heads, but that after a flip, but before looking, we can't say that there's a 50% chance that it is heads. I think that we can say that. — Michael

Your 50% applies to your uncertainty about the state of the coin. You flip a coin it has a 50% chance of T or H, after that lands that part is done. Without seeing the coin you guess there is 50% chance it is heads. That 50% is about your guess. One is subjective the other is objective, but I agree with your general direction.

This is not well-defined. It needs re-stating to make it unambiguous. — andrewk

The last thing that really needs to be addressed is this. @andrewk was correct that X needs to be defined, he was just wrong in how that needs to be done. It needs to be defined with notation with an index to represent the unknown limit and unknown distribution.

If you want your equations to be complete @Srap Tasmaner you need to address Andrewk's standards here. I don't want to sort the notation myself, but you seem to enjoy that aspect.

This may help:

https://www.encyclopediaofmath.org/index.php/Random_variable

I tried very hard to avoid replying to that behavior, but you wouldn't let me. — JeffJo

The only control I have over you is what you allow me to have.

So that "observational data set" is the "experimental data set," isn't it? — JeffJo

I find it interesting that you didn't pick up on what happened there.

You are removing it from its context to make it look bad — JeffJo

Just following your lead.

Already did that. It was not that hard.