With all those pages of text and way more than necessary of technical terms, jeffjo is still not convinced. I would like to try but first i must make sure that he is reading this message or my attempt would be in vain.

All those explanation against jeffjo, even correct, but does not hit the nail in the head and fail to convince him. ]]>

It will surprise many people, but nobody is using the correct wording of the problem. They are including elements that Adam Elga introduced in order to explain his thirder solution simply. The two wordings have the same answer, of course, but the objections Halfers raise to his solution all relate to the parts he introduced.

Here is the actual wording from Elga’s paper, with two edits that remove irrelevant information: “Some researchers are going to put you to sleep. During the [time] 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 [awake], to what degree ought you believe that the outcome of the coin toss is Heads?”

In the original wording, my “[time]” was two days. But it didn’t actiually say the wakings would occur on different days, and no order was specified. My “[awake]” was “first wakened,” which is a bit ambiguous. Since no order is specified, it can’t mean “Monday, but not Tuesday.” And since Elga’s solution involved telling SB information after her first answer, it seems obvious that it means “before podssibly learning something.”

There is another way of implementing this, where the “new information” becomes obvious, and the Halfer’s methods work against them.

Once SB is asleep, flip two coins, a Quarter and a Nickel. If either is showing Tails, (1) wake her, (2) ask her the question, and after she answers, (3) put her back to sleep with amnesia. Then, TURN THE NICKEL OVER, and repeat the same three steps. The question is “to what degree ought you believe that the Quarter is showing Heads?”

Note that the Quarter never changes, so what it is currently showing is the same as the outcome of its flip. But even though the Nickel might not be showing the what its flip-outcome was, the prior distribution for the two coins is found by Halfer techniques. Each of {HH, HT, TH, TT}, where the Quarter is listed first, has a probability of 1/4.

But the “new information” that SB has is that HH is eliminated. She wouldn’t be awake, and facing the question, it both coins afre currently showing Heads. The remaining three combonations each update to a conditional probability of 1/3. Since only one of them has the Quarter showing Heads, the answer to the question is 1/3.

]]>Some argue that Beauty, upon awakening, learns no new information, and therefore the priori probabilities of the initial coin flip are valid. While it is true that B learns no new information, that is NOT what determines the posterior probabilities; they are changed by the simple act of waking up – thus entering phase two of the experiment, the first one being the coin flip. For example, consider the same rules but for flipping the coin a second time on tails to determine whether Beauty should be awoken on Monday or Tuesday. Now the answer is heads with 1/2. The equations remain the same, but Pr(C=h|D=mon) evaluates to 2/3 instead of 1/2.

(1) and (2) work for all possible scenarios with respect to awakening Beauty. For Pr(C=h|D=mon):=1 however, signifying Beauty is never awoken on Monday for tails, they evaluate to x=y, meaning Monday and heads are synonymous, thus C taking on its priori probabilities, 1/2.

]]>Let C be the coin flip {h,t}, and D the day {mon,tue}. Then

(1) y := Pr(C=h) = Pr(C=h|D=mon)*Pr(D=mon) + Pr(C=h|D=tue)*Pr(D=tue) = 1/2*x + 0*(1-x) = x/2

(2) x := Pr(D=mon) = Pr(D=mon|C=h)*y + Pr(D=mon|C=t)*Pr(C=t) = 1*y + 1/2*(1-y) = x/4 + 1/2

(1) and (2) defines an equation system in x and y with solution 2/3 and 1/3 respectively. QED.

“:=” means “defined as” – introducing a definition.

Thus, there is no need for convoluted explanations betting on outcomes, or sidetracking with modified problems. All the information needed for solving the problem at hand is readily available in the problem description.

]]>First, consider a meaningless change: SB will be assigned, but not told, a random coin result C from the set {Heads,Tails}, and a random day D from the set {Monday,Tuesday}. She will be put to sleep on Sunday night, and the coin will be flipped. She will be wakened at least once, and maybe twice, during the next two days. She will he left asleep only on day D, and only if the actual coin flip was C. Anytime she is wakened, she is asked for her assessment of the probability that the coin result was C.

The change is meaningless because it is the original question when D=Tuesday and C=Heads, and the other combinations are symmetric variations that must have the same answer.

Then, use four volunteers and only one coin. Accomplish the random assignments mentioned earlier by randomly distributing the four possible combinations of {C,D} among them. On each day, three of the volunteers will be wakened, but only one of those three will have been assigned the C that matches the actual flip. So each can confidently answer 1/3 to the question asked.

]]>In my previous post I have presented the probabilities for both definitions of the event Red. It is clear that we both agree that in case we define Red as the event “SB will be led into a red room in this line of history” (call it Red1) P(Red1)= 1/2 and that if we define the event Red as “SB observes (at her current awakening I may add) that she is led into a red room” (call it Red2), that P(Red2)=1/4. My argument is that you can’t define the events unless you have already defined a random experiment whose outcomes constitute the sample space these events belong to. This is why I have described two random experiments, one corresponding to the first interpretation of event Red (Red1) and the other to the second interpretation (Red2). The subtle point is (and to my opinion the main reason for all the controversy around SB problem) that although the first random experiment actually takes place (there is a coin flip whose outcome is random), the second random experiment never actually takes place. It is used by SB to model her uncertainty on which stage of the actual experiment she is at her current awakening. So it is perfectly consistent to model her ignorance upon her current state as if the stage she is at has been randomly selected and assign P(Red)=1/4 and P(Blue)=3/4 for her current awakening. However, she knows that such random selection never actually takes place. Thus, if she is actually led on a Blue room she cannot update to P(Heads|Blue2)=1/3 but she still can use P(Heads|Blue1)=1/2.

Best regards,

Yannis ]]>

SB has to base her assessment on what she sees, not what fortune tellers may know. She only sees that she is led into a red room, not what may happen on another day.

]]>>> that event Red, what is P(Red)?

> P(Red)=P(Heads)=1/2

Incorrect. In three out of four possible situations, she will be led into a blue room. In one, she will be led into a red room. So P(Red)=1/4.

What you ignore, is that there are two different days that occur after a “heads” result. You consider “Room=Red” to be true on some Mondays when she is led into a blue room. Yes, she will be led into a red room the following day, but when she observes “Room=Blue,” it is not true FOR THAT OBSERVATION regardless of whether it is true on another day.

Similarly, in the original experiment, she will be awakened in one of three possible situations. When she finds herself awake, each is equally likely to be her current situation. Your error is considering a different situation to be her current situation, just because it is guaranteed to happen. The error is that it ISN’T happening.

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