JeffJo

What kind of misrepresentation you are talking about? You have literally just claimed that there is no way to know whether a particular probabilistic model correctly represents the system.

Since what I said was that probability theory makes no restrictions on how to assign values, but that you have to make assignments that are reasonable based on things like the Principle of Indifference, this would be an example of misrepresentation.

If what you are suggesting were true, then the correct answer to "What is a probability of a fair coin toss, about which you know nothing else, to come Heads?", would be: "It is either 0 or 1, but we can't tell.

What? No, absolutely

My criteria are what is required for the Laws of Probability to apply. There are no criteria for being a "true probability statement," whatever it is that you think that means.

There is only one criterion - it is more of a guideline than a rule - for how to assign values to probabilities. It's called he Principle of Indifference. If you have a set of events, and no reason to think that any one of them is more, or less, likely than any of the other events in the set (this is what "indifferent" means), then you should assign the same probability to each. You don't have to, of course, but that... (read more)

Were it true, then the correct answer to "What is a probability of a fair coin toss, about which you know nothing else, to come Heads?", would be: "Any real number between zero and one".

Yes, you can make a valid probability space where the probability of Heads is any such number. What you can't do, is say that probability space accurately represents the system.

Probability Theory only determines the requirements for such a space. It makes no mention of how you should satisfy them. And I assumed you would know the difference, and not try to misrepresent it.

I have no idea how you managed to come to the conclusion that I don't understand

Here is a  similar problem based on the same Mathematics. I call it Camp Sleeping Beauty. The same sleep and Amnesia details will be used, but at a week-long camp. On the arrival day, Sunday, you will be told all these details.

1. There are six "camp" activities, and you will partake of a random one on each day: Archery, Boating, Canoeing, Dodge Ball, Exercising, and Free Time.
2. The random part is that, after you are put to sleep on Sunday, a six-sided die will be rolled. But before you are put to sleep, you will help to randomly populate a six-by-six Camp Calendar with these activities. The board has one column for each of

Well, it is true, and your confusion demonstrates what you don't want to understand about probability. It is not about what what can be shown, deterministically, to be true when we have perfect knowledge of a system. It is about the different outcomes that could be true when your knowledge is imperfect.

Consider your "opaque box" problem. You are not "assigning non-zero probability to a false statement," you are assigning it to a possibility that could be true based on the incomplete knowledge you have. This is what probability is supposed to be - a measure of your lack of knowledge about a result, not a measure of absolute truth. Even if there... (read more)

You hit one point on the head: "it’s really just a math problem." But what that means is that it is not a philosophy problem. It was originally posed as a problem in philosophy, but the only thing such considerations do is obfuscate the math problem. Without justification. Just the need to put it in the realm of philosophy to match the original intent.

Consider a variation of the problem, that isn't a variation at all, but I'm sure will get a response as to why it doesn't fit in with a desired philosophical solution. I know this, because it has happened before. Use the same details with four volunteers, but one coin,... (read more)

These problems are actually variations of one of the oldest "probability paradoxes" ever. And I put that in quotes, because in 1889 when Joseph Bertrand published it, "Bertrand's Box Paradox" meant how he proved that one proposed answer could not be right, because it produced a contradiction.

Here is that paradox, applied to Problem #1. With a slight modification that changes nothing except using a complimentary probability. In each question, what is the probability that I have a boy and a girl?

• I have two children, at least one of whom is a boy.
• I have two children, at least one of whom is a girl.
• I have two children, and have written the gender of

Contrary to what too many want to believe, probability theory does not define what "the probability" is. It only defines these (simplified) rules that the values must adhere to:

1. Every probability is greater than, or equal to, zero.
2. The probability of the union of two distinct outcomes A and B is Pr(A)+Pr(B).
3. The probability of the universal event (all possible outcomes) is 1.

Let A="googolth digit of pi is odd", and B="googolth digit of pi is even." These required properties only guarantee that Pr(A)+Pr(B)=1, and that each is a non-zero number. We only "intuitively" say that Pr(A)=Pr(B)=0.5 because we have no reason to state otherwise. That is, we can't assert that Pr(A)>Pr(B) or Pr(A)<Pr(B), so we... (read more)

I don't think I understand what you've written here. It's indeed possible that the card is not Club when it's Spade. As a matter of fact, it's the only possibility, because the card can't be both Spade and a Club.

There are four different days when SB could be awakened. On three of them, she would not have been awakened if the card was a club. This makes it more likely that the card is a club. This really is very simple probability. If you have difficulty with it, wake her every day. But in the situations where she was left asleep before, wake her and ask her for the probability that the... (read 818 more words →)

Here's a new problem that requires the same solution methodology as Sleeping Beauty.

It uses the same sleep and amnesia drugs. After SB is put to sleep on Sunday Night, a card is drawn at random from a standard deck of 52 playing cards.

On Monday, SB is awakened, interviewed, and put back to sleep with amnesia.

On Tuesday, if the card is a Spade, a Heart, or a Diamond - but not if it is a Club - SB is awakened, interviewed, and put back to sleep with amnesia.

On Wednesday, if the card is a Spade or a Heart - but not if it is a Diamond or a Club - SB is awakened,... (read more)