I have flipped a coin and know the result. I will let you bet on it with a 2:1 payout if you win. However, I decide how much you have to bet, and only after you tell me your guess.

Should you bet? — Michael

I don't understand how this is supposed to be comparable or relevant to the envelope paradox. Are you the person opening the other envelope or the person organizing the test?

Both people know the amount in their envelope. Let's say they can both choose the same envelope but can't communicate with each other. What should they choose? Always choosing the envelope of the other person can't be beneficial unless A+B≠B+A.

What if there are two people and each time both are randomly given an envelope. Is it then beneficial for both to switch the envelopes? So each of them gets more than half of the money on average.

What if these two people then put the money back together and distribute it in new envelopes? Do they by switching the envelopes again keep infinitely increasing the amount of money?

1. We pick an envelope at random
2. There's a 50% chance that my envelope is the X envelope and a 50% chance that my envelope is the 2X envelope.
3. I open my envelope and see £10
4. From 2 and 3, there's a 50% chance that my £10 envelope is the X envelope and a 50% chance that my £10 envelope is the 2X envelope. — Michael

False, learning the amount changes the odds. There is no way to choose those two amounts of money in such a way that any amount is as likely to be X or 2X.

The intuitive solution is that the bigger the amount in your envelope is, the more likely it is to be the one with 2X.

This is correct when waking up just once, so why not also when possibly waking up twice? — Michael

Because when you are woken up more than twice more awakenings are caused by throwing tails.

The Mondays have to be as likely. Let's say the coin is thrown on Monday evening instead. It's clear now that the coin flip can't affect that since it happens after the test.

The OP is referring to the fact that congressional legislation is nearly always (something like >90% of the time) in line with what the very wealthy wish. — MindForged

It's not just the wealthy that have power, it's not that simple. The government is also an entity in its own that has power over the rich people, and it's connected to many other entities like CIA or army that don't have so clear ties to the class economics. In fact, these have to do with the topics in respect to which the common folk are completely indifferent that I mentioned. The rich have a big role in making USA the place it is, but that's more building up anger, unhappiness, dissatisfaction and conflicts than fear or culture. For example the countless human rights violations made by CIA don't really affect average people, but they build up certain mental images of what those organizations and by associations USA as a whole are. These mental images create the collective thoughts that America is not safe etc.

Notice how you've decided (correctly) not to consider Heads + Tuesday and argue that in my example there's a 1/3 chance of it being heads. Why the inconsistency? — Michael

Because unlike in the Sleeping Beauty problem, in your example if there's x probability of going through any day you can draw the conclusion there's x probability of it being that day. The heads + Tuesday option is still there, but because of the rules of your variation the odds of going though any outcomes with a specific coin flip need to add up to 50%.

Well, the point is that the government is serving interests other than that of the common folk. — Posty McPostface

That's too deep underlying imo. It's the actions that matter more, not their reasons, especially if we're looking at the reasons for the fear of citizens. That's like a reason of a reason but there's no direct connection. How deep are we going to dig? Let's ask why the government has other interests. Maybe because of the politicians making decisions being too far away from the common folk so that they can't relate to them anymore. Why is this? I might say because there's too much power in USA so that they can do decisions that don't directly relate to the people the government should by its existence serve. USA's rise to a superpower is a complicated issue but I'll pick two historical reasons: the arms race against USSR and maybe the events after WWII. WWII was caused by Nazis, who raised as an answer to the threat of communism, and communism also caused the existence of the Soviet Union.

So are the Americans afraid of their government because of the industrial revolution leading to bad working and living conditions for the working class, which led to Karl Marx coming up with communism, because of which Nazi Germany and the Soviet Union both came into existence, WWII happened, USA won it, got an economic boost, started competing against the Soviet Union, became a superpower, and all this lead to the government of USA having bigger concerns than their citizens, because of which their interests weren't the top priority, which led to USA mistreating their citizens?

If we continue that long enough we get to the boundary conditions of the Universe together with deterministic laws of physics, or God's plan, or the chaoticity and meaninglessness of our existence and the lack of any real reason, or whatever. That big metaphysical answer behind everything is technically correct but I don't find much meaning in it. So I just prefer cutting the reasoning after the most straightforward explanation according to Occam's Razor.

Sleeping Beauty knows she has 1/2 chance of going through any possibility. You have 1/2 chance of going through Monday+heads on 1/4 chance of going through each tails outcome. Those situations are not comparable.

I’m saying that if it’s heads then only I’m asked and if it’s tails then I’m asked and 1,000 other people are asked. — Michael

That's not how it works, it needs to be a random person chosen for the situation to be comparable. The Sleeping Beauties at different scenarios are comparable to separate persons because of the amnesia, except arguably the Monday versions, who however don't have the knowledge it's Monday, unlike you, who always has the knowledge you are you.

You're not guaranteed to be asked if random people are chosen for the questioning either.

So it's not a false comparison because the answer is still 1/2?

Actually it's almost exactly 50%. But what you're saying is that if we choose random people to the test for each questioning instead of waking up the same person, then the answer is 1/3?

If I throw heads I choose a random person to question. If I throw tails I choose 7 599 999 999 people to question. You find yourself to be a part of the experiment, and don't have any knowledge on whether other people were chosen.

What are the odds you're a part of the experiment because heads was thrown?

But that doesn't mean that there was a 1/3 chance of getting a red ball. — Michael

This is not what the question is about. The question is whether there's 1/3 chance of being questioned as a result of getting a red ball.

This is why I suggested the alternative experiment where we don't talk about days at all and just say that if it's heads then we'll wake her once (and then end the experiment) and if it's tails then we'll wake her twice (and then end the experiment). There aren't four equally probable states in the experiment. — Michael

There's still the fourth state of it being the time you'd be awakened the second time and heads was thrown.

I think the issue is that the goals of "the government" are not aligned with the interest of the common folk. I call this the alignment problem. — Posty McPostface

I think that's over-thinking it. The government is dangerous and potentially harmful to the citizens, and while the actions reflecting that are not goals for the people I don't really see the insight looking at the alignments has to offer.

I never stated belief in the claim to be false. I originally stated, " I do not believe a God exists ". This is NOT the same as, " I believe NO God exists " - These are two different claims. Or you can look at it this way.

Claim: God exists
Me: I don't accept that claim as true. — chatterbears

My reply was to the example with invisible pet dragon. In informal language as it's commonly used the negation of the phrasal verb "to believe someone" means the same as thinking they're lying, and not that one doesn't hold their beliefs.

I'd suggest researching a little more on the burden of proof — chatterbears

My analysis on the burden of proof is completely valid on my stated premises about what the sentence "I don't believe you" is interpreted to mean.

What are you talking about? Explain to me how the burden of proof rests on the person who does not believe the person making the claim. You seem to not understand the burden of proof and where it rests. It rests on the person making the claim, not the person who doesn't accept the claim as true. — chatterbears

Depends. By "does not believe the person making the claim" do you mean the same as "believing the claim to be false"? If so, there is a burden of proof on that belief, as a new claim is made and each claim has a burden of proof on it. But if you mean the person questioning the claim has no belief either way regarding the claim, you're correct, there's no burden of proof on them.

If you claim that you are an atheist, you surrender the bases on which to make claims about the nature of god. — Txastopher

I don't see how so. You can refer to claims of other people or make speculations perfectly well as an atheist.

OP claims that god is wrong — Txastopher

No, just that he thinks so. He only claims they would(/could) disagree. The following

This is a claim about the nature of god — Txastopher

is only true with objective morality, so the claim is more about the nature of morality if anything. You could argue that if God was objectively right they'd convince you so but that feels far-fetched to be honest - then you couldn't really say you'd be following the orders out of your free will.

Anyway, I can search for statistics in some cherry picking manner to support my point of people being paranoid and fearful of the government in the US. — Posty McPostface

The term "paranoid" implies the fear is irrational, so you'd need to compare that data to actual dangers. I'm not paranoid for wanting to own a gun in the Middle East, and neither are the Americans afraid for no reason (although the fear of the conspiracy theorists is irrationally targeted towards stupid beliefs, sorry to put it so bluntly). I refer to the link I posted before, to this ridiculously long list.

The problem is that you can't take an atheist position and make claims about the nature of the deity. — Txastopher

I'm claiming one doesn't need to. You can choose to not follow someone's commands without making any claim. Do I believe you have a beard? No. Do I believe you don't have a beard? No. Will I go to your church if you tell me to? No. Does not mean I have to make a claim about your beard.

Ah, so now you're claiming that whatever god had to say, it may or may not be correct. So this god that you claim not to believe in has, nevertheless, certain qualities that you are sure about. What a mess! — Txastopher

How can certainty about god or any of their qualities be concluded from not following their commands?

Obviously, you exclude yourself from this group since, as you state above, you wouldn't necessarily follow his commands — Txastopher

Which could perfectly well be done while being aware of what those commands were.

My claim is, "I do not believe a God exists." AKA "I am not convinced that God exists." - If I told you I owned an invisible pet dragon, and you said you don't believe me, does the burden of proof suddenly rest with you? No. — chatterbears

Yes it does. It's simultaneously on both of you. Your claim offers no new information so we simply don't know whether you have an invisible pet dragon. Drawing conclusions from the existence of burden of proof or from that someone, despite the burden of proof being on them, doesn't provide any proof, is argumentum ad ignorantiam.

What an awfully long way to say "Americans are obsessed with guns because they're afraid that someone they're oppressing will become dangerous".

One phenomenon that strikes me as strange, and which is not even a recent thing is the amount of paranoia and fear towards our own government. — Posty McPostface

Maybe because they should be afraid.

en.wikipedia.org/wiki/Unethical_human_experimentation_in_the_United_States

In the original problem adding any outcomes, like heads and Tuesday, doesn't affect the other odds in respect to each other. If some specific interview is as likely as another one (like tails and Monday and heads and Monday) we can stuff in new outcomes and those relations don't change.

In the coin flip example to get into any outcome you need to take away some outcome from which to get to the new outcome. Like heads is twice as likely as tails + heads, but heads + tails removes heads because you have to proceed with a new flip.

No, if you consider heads + tails the chances go to 0,5. Eliminating that possibility results in 0,75.

I'm saying that there is an eliminated outcome – heads + heads – and yet the probability is still 0.75. — Michael

With the outcome added there are certain odds, and when that is modified by eliminating that one possibility the odds are changed in a way that somehow makes sense. It's just another way to look at the problem, like how in the Sleeping Beauty problem the chances are 2/3 whether one thinks about the eliminated extra possibility or not. Thinking about it doesn't change the situation but coming to the same conclusion through eliminating it can be use to confirm the solution.

But as we disagree about what conclusion is reached by looking at the problem that way I don't think we can achieve an agreement through it.

Where has Tuesday come from? It wasn't mentioned at all in the experiment I described here. — Michael

Because that's the original scenario where it's already eliminated.

Then you might as well say that we've eliminated the trivial scenario of heads + heads in this experiment. — Michael

Completely possible to do so.

Heads + heads utterly irrelevant and doesn't change the probability at all. — Michael

You mean modifying that scenario so that it includes heads+heads doesn't change the probabilities in it?

As I said, Tuesday and heads. From the trivial scenario where she's woken up twice regardless of the coin flip.

There's nothing to eliminate. This is it. — Michael

Because that's where we get to after we eliminate the one possibility from the aforementioned case.

We flip a coin. If it's heads then we wake Mary once. If it's tails then we wake her twice.

What is the probability that it's her first awakening? 2/3, because two of the three outcomes are first awakenings? Or 3/4 because the probability is 0.5 + (0.5 * 0.5)? — Michael

You forgot to weigh the calculations for the potential two wakings. Run the test 100 times, she'll wake up the first time 100 times out of 150.

Not from there. That's where you get when you first take the trivial case where she's woken twice either way.

That's like saying you can't eliminate 1 from the set {1} to get { } because there's nothing in { } to eliminate.

and can ignore your "there's no 'eliminated" outcome" objection — Michael

No, we can't. Eliminating that possibility diminishes the chances of heads being the correct guess.

We flip a coin. If it's heads then the result stands. If it's tails then we flip again and the new result stands.

What is the probability that it's heads? 2/3, because two of the three outcomes are heads? Or 3/4 because the probability is 0.5 + (0.5 * 0.5)? — Michael

3/4 but the situation is completely different. No difference between observers. Outcomes lead to same amounts of inquiries.

Now this example right here:

I toss a coin and ask you what I got, but if it's heads I toss it again and if it's tails that round isn't played. What are you guessing when I ask you what I got? — BlueBanana

This one's cool because it too can be calculated quite simply. 1/2 chances of tails and (1/2)*(1/2)=1/4 chances of both no question and heads.

Now what happens if you repeat the test? The Sleeping Beauty problem.

That's irrelevant and doesn't have anything to do with the probability. We can change the scenario slightly to:

If it's heads then we wake her once. If it's tails then we wake her twice.

What's her credence that it's heads? — Michael

Then it's likelier the waking up was caused by flipping tails. As I said,

there are two guesses for each flip of a tails
— Michael

Because of which each guess is more likely to have been caused by tails. — BlueBanana

Heads and Tuesday.

It wins more times — Michael

This is just another way of saying it's more likely to win.

there are two guesses for each flip of a tails — Michael

Because of which each guess is more likely to have been caused by tails.

We flip a coin. If it's heads then the result stands. If it's tails then we flip again and the new result stands.

What is the probability that it's heads? 2/3, because two of the three outcomes are heads? Or 3/4 because the probability is 0.5 + (0.5 * 0.5)?

This seems to be the crux of the disagreement. — Michael

Completely different situation. There's no "eliminated" outcome. There's no difference between an outside observer and the subject.

It is in this context. With each guess there's the same bet, same amount of money, same chances of winning. So if guessing tails doesn't win more times, why is it profitable?

How about this: I toss a coin and ask you what I got, but if it's heads I toss it again and if it's tails that round isn't played. What are you guessing when I ask you what I got?

The above reasoning only works if --- — Michael

It works now. It explains why guessing tails is profitable.