JeffJo

Q1: Is the answer the same if SB is left asleep on H+Mon, but wakened on H+Tue?

Q2: Is the answer the same if the day SB is left asleep, H+Mon or H+Tue, is determined by a second coin flip?

Q3:  On Sunday Night, flip two coins. Call them C1 and C2. Coin C1  is the one SB is asked about, and C2 is the second on in Q2. So, on Monday wake SB if either coin is showing Tails. On Monday Night, turn C2 over to show the opposite side. And on Tuesday, again wake SB if either coin is showing Tails. Whenever awakened, what should SB answer about coin C1? But before ans... (read more)

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

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 ... (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

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

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... (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 bef... (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 tw

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 h... (read more)

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 awakene... (read more)

Yes! I'm so glad you finally got it! And the fact that you simply needed to remind yourself of the foundations of probability theory validates my suspicion that it's indeed the solution for the problem.

Too bad you refuse to "get it." I thought these details were too basic to go into:

A probability experiment is a repeatable process that produces one or more unpredictable result(s). I don't think we need to go beyond coin flips and die rolls here. But probability experiment refers to the process itself, not an iteration of it. All of those things I defined b... (read more)

A Lesson in Probability for Ape in the Coat

First, some definitions. A measure in Probability is a state property of the result of a probability experiment, where exactly one value applies to each result. Technically, the values should be numbers so that you can do things like calculate expected values. That isn't so important here; but if you really object, you can assign numbers to other kinds of values, like 1=Red, 2=Orange, etc.

An observation (my term) is a set of one or more measure values. An outcome is an observation that discriminates a result suffi... (read more)

The link I use to get here only loads the comments, so I didn't find the "Effects of Amnesia" section until just now. Editing it:

"But in my two-coin case, the subject is well aware about the setting of the experiment. She knows that her awakening was based on the current state of the coins. It is derived from, but not necessarily the same as, the result of flipping them. She only knows that this wakening was based on their current state, not a state that either precedes or follows from another. And her memory loss prevents her from making any connection between the two. As a good Bayesian, she has to use only the relevant available information that can be applied to the current state."

Let it be not two different days but two different half-hour intervals.  Or even two milliseconds - this doesn't change the core of the issue that sequential events are not mutually exclusive.

OUTCOME: A measurable result of a random experiment.

SAMPLE SPACE: a set of exhaustive, mutually exclusive outcomes of a random experiment.

EVENT: Any subset of the sample space of a random experiment.

INDEPENDENT EVENTS: If A and B are events from the same sample space, and the occurrence of event A does not affect the chances of the occurrence of event B, then A a... (read more)

And as I’ve tried to get across, if the two versions are truly isomorphic, and also have faults, one should be able to identify those faults in either one without translating them to the other. But if those faults turn out to depend on a false analysis specific to one, you won’t find them in the other.

The Two Coin version is about what happens on one day. Unlike the Always-Monday-Tails-Tuesday version, the subject can infer no information about coin C1 on another day, which is the mechanism for fault in that version. Each day, in the "world" of the subject... (read more)

This is the Sleeping Beauty Problem:

"Some researchers are going to put you to sleep. During the two days that your sleep will last, they will briefly wake you up either once or twice, depending on the toss of a fair coin (Heads: once; Tails: twice). After each waking, they will put you to back to sleep with a drug that makes you forget that waking. When you are first awakened, to what degree ought you believe that the outcome of the coin toss is Heads?"

Unfortunately, it doesn't describe how to implement the wakings. Adam Elga tried to implement by adding s... (read more)

This paper starts out with a misrepresentation. "As a reminder, this is the Sleeping Beauty problem:"... and then it proceeds to describe the problem as Adam Elga modified it to enable his thirder solution. The actual problem that Elga presented was:

Some researchers are going to put you to sleep. During the two days[1] that your sleep will last, they will briefly wake you up either once or twice, depending on the toss of a fair coin (Heads: once; Tails: twice). After each waking, they will put you to back to sleep with a drug that makes you forget that wak

My problem setup is an exact implementation of the problem Elga asked. Elga's adds some detail that does not affect the answer, but has created more than two decades of controversy.

The answer of 1/3.

I keep repeating, because you keep misunderstanding how my example is very different than yours.

In yours, there is one "sampling" of the balls (that is, a check on the outcome and a query about it). This one sampling is done only after two opportunities to put a ball into box have occurred. The probability you ask about depends on what happened in both. Amnesia is irrelevant. She is asked just once.

In mine, there are two "samplings." The probability in each is completely independent of the other. Amnesia is important to maintain the independence.

SPECIFICAL... (read more)