
#19
Nov713, 03:57 PM

P: 503

Hold on, I may be confused about the calculation of expectation value, or probability, or both...
On the one hand, after opening one envelop and finding $100, the initial possible configurations of money in the envelops might be interpreted to have must been: ($50, $100) or ($100, $200) based on the phrase, "...one has twice the money in it as the other..." then the expectation value of the first is ($50+$100)/2 = $75 and the second is ($100+$200)/2 = $150 so if choosing either one had a p=.5 then the expectation value of having chosen either envelop was: ($75=$150)/2 = $112.50 If this was a game in which after each selection one was allowed to "buy" the other envelop with the money you got from your selection, your strategy would be to always do so... kind of a paradox if you see the initial selection as random. One the other hand, there may be a problem with assuming too much about the phrase "one has twice the money in it as the other"... That relationship applies between two real extant values of which only one is known. That relationship does not necessarily extend between the known value and a third hypothetical value based on a counterfactual hypothesis... For example, if the initial condition was this: ($100, $200) that satisfies the "one has twice the money in it as the other" stipulation. But when one reveals the $100, one does not know if the $100 is the lower or higher value. Extending the possibility to the case to ($50, $100) or ($100, $200) seems unjustified using the "twice" stipulation because the same initial condition might have been stipulated that: "one has $100 more money in it as the other" in which the ($100, $200) case satisfies the stipulation, but one of the hypothetical cases would be: ($0, $100) which is different from ($50, $100). The "twice" stipulation may be replaced by others that the initial condition satisfies but which would generate a whole host of different hypothetical cases. I guess what I'm thinking is that if a particular stipulation is only one of many that achieve the same relation, what is the basis for extending that particular relation to yield the hypothetical case values? 



#20
Nov713, 04:56 PM

PF Gold
P: 3,075

The connection with the Doomsday Argument is that we cannot assume we have a 95% chance of being in the last 95% of humans, if we also know that our birth number is about 10 billion. There is unknown information about how long intelligent civilizations last that can introduce correlations between birth number and probability of being in the last 95%, and simply not knowing those correlations does not justify asserting we will get a correct result by assuming there are none. 



#21
Nov713, 08:13 PM

P: 503

Thanks, that makes sense.
I looked at the Doomsday argument at Wikipedia, and it does not make sense. Wiki first states, "...it says that supposing the humans alive today are in a random place in the whole human history timeline, chances are we are about halfway through it." This seems to imply that on a uniformly distributed interval of [0, 1] that the most likely random place is .5 half way through it. That might be true if the distribution was like a normal distribution with a peaked center and trailing tails, but wouldn't a uniform distribution would offer the same probability to all values in the interval and not favor the half way point? Then Wiki states, "...suggests that humans are equally likely (along with the other N − 1 humans) to find themselves at any position n of the total population N, so humans assume that our fractional position f = n/N is uniformly distributed on the interval [0, 1]..." This seems to imply that on a uniformly distributed interval of [0, 1] that the most likely random place is not .5 half way through it, that all points are equally probable  different from the first statement. It seems to me that if the selection is of a random time within the interval, then all times are equally probable and the attaching of a human to that time is incidental or independent; but if it is the selection of a human that is being done from within the total historical population in the interval, then the corresponding time location for that human is going to be more likely in the population dense direction of the time interval. Maybe the first state is just very clumsy and trying to imply the second statement, but if that is so, why go with the "half way" conclusion if the conclusion of the second is "uniform"? So it goes back to "...supposing the humans alive today are in a random place in the whole human history timeline..."; which is being selected? It is not clear whether the selection is of a place in the time line or a group of humans in the time line population. In any case, it looks to me like the logical error happens when moving from the random position of a human in the population history to mapping that human into the time line. The graph clearly shows that can't be done because the selection of the human from the population occurs as if the population is mapped to a number line... n of N where these are integers. But on the historical time line the population density stacks up toward the future end of the interval. The confidence interval calculations for the position in the population when spread out evenly as a number line of integers from 0 to N can't be applied to the time line interval where the population accumulation is not in an even line, but all stacked or folded or compounded to make the increasing density... What makes it worse for someone who is not used to some of this is if you follow the figures and construct a similar graph with probability on the vertical axis (going from 0 to 1) and time line period on the horizontal axis as the interval [0, 1] going from 0 to 1... then a uniform probability is going to represent a horizontal line at p=.5 which may be misunderstood as the basis for the "half way through it" remark, since .5 and half way and 50% are all similar, and for any instance on the time line axis the value is going to give .5 



#22
Nov713, 08:31 PM

PF Gold
P: 3,075





#23
Nov813, 12:11 AM

P: 503

So Wiki is incorrect when it states that, "f is uniformly distributed on (0, 1] even after learning of the absolute position n." ?
I'm assuming the Wiki switch from [0, 1] to (0, 1] is just a typo and not sleight of hand... 



#24
Nov813, 01:58 AM

PF Gold
P: 3,075





#25
Nov813, 08:59 AM

Mentor
P: 10,853

The first 5% and the last 5% would always get it wrong. Using the same argument, humans born in 1970 could have concluded that they were not in the first 5% born after the moon landing  and would have been wrong. All of them. A proper analysis would need Bayesian statistics here, but we have no idea how a proper prior would look like*, so we cannot make probability calculations based on the number of humans that lived on earth so far. *an extreme example: "once a species reaches the technology level of spaceflight, it will colonize a significant fraction of a galaxy with a probability >90%" is a possible scenario. In this case, we would be within the first 5% with a high probability. 



#26
Nov813, 11:00 AM

PF Gold
P: 3,075





#27
Nov1113, 01:39 PM

P: 85

Only if an asteroid causes another planet's orbit to change, pulling Earth out of its orbit and possibly plunging Earth into the Sun, or hitting the Earth itself and altering Earth's orbit to the same effect.



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