**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 sufficiently for the purposes of the random experiment. An experiment’s *sample space *is a set of all distinct outcomes that are possible for that experiment; it is often represented as Ω.

There is no single way to define outcomes, and so no single sample space, for any experiment. For example, if you roll two dice, the sample space could be 36 unordered pairs of numbers, 21 ordered pairs, or eleven sums. But what “sufficient” means is that every observation you intend to make is its own outcome in the sample space. You could divide the results of rolling a single die into {Odd, Even}, but that won’t be helpful if you intend to observe “Prime” and “Composite.”

An event is any subset of the sample space, not a specific result. (You, Ape in the Coat, confuse it with either an observation, or an outcome; it isn’t clear which.) We could also talk about the event space F and the corresponding probability space P, but those exact details are also not important here. Except that P corresponds to F, not Ω. When I talk about the probability of an outcome, I mean the solitary event containing just that outcome.

Finally, conditional probability is used when an observation divides the sample space into two sets: one where every outcome is 100% consistent with the observation—that is, no un-observed measure makes it inconsistent—and its compliment where no outcome is consistent with the observation. (This is where people go wrong in problems like Monty Hall; the outcome where Monty opens Door 3 to show a goat is not the same as the observation that there is a goat behind Door 3).

With these definitions, we can see that the SB Problem is one random experiment with a single result. That result is observed twice (yes, it is; remaining asleep is an observation of a result that we never make use of, so awareness as it occurs is irrelevant). What you call “sequential events” are these two separate observations of the same result. That’s why you want to treat them as dependent, because they are the same result. Just looked at a different way.

And the effect of the amnesia drug is that any information learned in “the other” observation does not, and * can not*, influence SB’s belief in Heads based on the current assessment. Knowing

*that*an observation is made is not the same thing as knowing

*what*that observation was.

Elga’s sample space(s) are controversial because he uses different measures for the two observations. So how they relate to each other seems ambiguous to some. He actually did it correctly, by further conditionalizing the result. But since his answer violated many people’s intuitions, they invented reasons to argue against how he got it.

My two-coin version does not have this problem. The sample space for the experiment is {HH1_HT2, HT1_HH2, TH1_TT2, TT1_TH2}. Each outcome has probability ^{1}⁄_{4}. The first observation establishes the condition as {HT1_HH2, TH1_TT2, TT1_TH2} and its complement as {HH1_HT2}. Conditional probability says the probability of {HT1_HH2} is ^{1}⁄_{3}. The second observation establishes the condition as {HH1_HT2, TH1_TT2, TT1_TH2} and its complement as {HT1_HH2}. Conditional probability says the probability of {HH1_HT2} is ^{1}⁄_{3}.

And the point is that it does not matter which observation corresponds to SB being awake, since the answer is ^{1}⁄_{3} regardless.

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

A

probability experimentis 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. Butprobability experimentrefers to the process itself, not an iteration of it. All of those things I defined before are properties of the experiment; the process. “Outcome” is any potential result of an iteration of that process, not the result itself. We can say that a resultbelongs to an event, even an event of just one outcome, but the result isnot the same thing asthat event. THE OBSERVATION IS NOT AN EVENT.For example, an event for a simple die roll could be EVEN={2,4,6}. If you roll a 2, that result is in this event. But it does not mean you rolled a 2, a 4, and a 6.

So, in …

… you are describing one iteration of a process that has an unpredictable result. A coin flip. Then you observe it twice, with amnesia in between. Each observation can have its own sample space—remember, experiments do not have just one sample space. But you can’t pick, say, half of the outcomes defined in one observation an half from the other, and use them to construct a sample space. That is what you describe here, by comparing what SB does on Monday, and on Tuesday, as if they are in the same event space.

The correct “effect of amnesia” is that you can’t relate either observation to the other. They each need to be assessed by a sample space that applies to that observation, without reference to another.

And BTW, what she observes on Monday may belong to an event, but it is not the same thing as the event.

A common way to avoid rebuttal is to cite two statements and make one ambiguous assertion about them, without support or specifying which you mean.

It is true that remaining asleep is a possible result of the experiment—that is, an outcome—since Tuesday exists whether or not SB is awake. What SB observes tells her that outcome is not consistent with her evidence. That’s an observation.

It is true that same the result (the coin flip) is observed twice; once on Monday, and once on Tuesday.

Or do you want to claim the calendar flips from Monday to Wednesday? That is, that Tuesday only exists is SB is awake? But if you still doubt this, wake SB on Tuesday but don’t ask her for her belief in Heads. Knowing the circumstances where you would not ask, she can then deduce that those circumstances do not exist. This is an observation.

What you do think makes it different than not waking her, since her evidence is the same when she is awake is the same?

No, thats how you try to misinterpret my version to fit your incorrect model. You use the term for Elga’s one-coin version as well. Strawman arguments are another avoidance technique.

Huh? What does “connect these pairs” mean to pairs that I already connected?

No, I am not. This is another strawman. I am describing how she knows that she is in either the first observation or the second. I am saying that I was able to construct a valid, and useful, sample space that applies symmetrically to both. I am saying that, since it is symmetric, it does not matter which she is in.

I only did this to allow you to include the “sequential” counterpart to each in a sample space that applies regardless of the day. The point is that “sequential” is meaningless.