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Is it possible for the earth to come off orbit and crash into the sun?

by andyM1998
Tags: crash, earth, orbit
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bahamagreen
#19
Nov7-13, 03:57 PM
P: 529
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?
Ken G
#20
Nov7-13, 04:56 PM
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Quote Quote by bahamagreen View Post
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.
Exactly, that's why we know the expectation value must be $100, so any calculation that gets a different result is incorrect.
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"...
The statement may be taken to be true. However, you are certainly right that an important part of the story are unseen correlations. In other words, we cannot hold that this is the only relevant information-- but all other relevant information is withheld from us. The same is true of the "Doomsday Argument"-- just because relevant information is unavailable to us, it does not mean we can assume it does not exist. One cannot always get away with that assumption when doing probability calculations, such as the claim that there is a 95% chance that we are not in the first 5% of all humans, given that we know our birth number is about 10 billion or so.
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?
Right, one does not know how to do that, which is why one gets an incorrect calculation of an expectation value if one makes certain unjustifiable assumptions. The only justifiable assumption is the symmetry principle that neither envelope is more likely to be worth more, so the other envelope must have a statistical value equal to what is revealed in the first. If you would buy the second envelope for any more than that, you will always lose money in the long run, no matter what system is used to stuff the envelopes.

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.
bahamagreen
#21
Nov7-13, 08:13 PM
P: 529
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
Ken G
#22
Nov7-13, 08:31 PM
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Quote Quote by bahamagreen View Post
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.
The selection isn't in time, it is in birth order. So it is normal to assume that all individuals are distributed evenly over birth order, that is, a randomly selected human from the full population is equally likely to have any birth number from 1 to N, where N is the total number of humans who ever live. But when we know that we are number 10 billion, say, we can no longer claim to be evenly distributed. That's the fallacy of mistaking an unknown correlation for a nonexistent correlation. It doesn't matter how the birth order maps into the time dimension, that's a separate issue that does not relate to the error in the argument (it brings in additional uncertainties, like assumptions about how the human population will wax or wane with time, but the argument is already wrong before it even gets to that point). It is just wrong to say that "since I don't know how my birth number correlates with where I stand in the total population, I may assume there is no correlation." The same error leads to the wrong expectation value for the second envelope.
bahamagreen
#23
Nov8-13, 12:11 AM
P: 529
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...
Ken G
#24
Nov8-13, 01:58 AM
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Quote Quote by bahamagreen View Post
So Wiki is incorrect when it states that, "f is uniformly distributed on (0, 1] even after learning of the absolute position n." ?
Yes, I have no idea on what basis they make that claim. For example, they also state (note N1 = 60 billion and N2=6,000 billion, and X is the 59+ billionth human):
Quote Quote by Wiki on Doomsday Argument
Now, if we assume that the number of humans who will ever be born equals N1, the probability that X is amongst the first 60 billion humans who have ever lived is of course 100%. However, if the number of humans who will ever be born equals N2, then the probability that X is amongst the first 60 billion humans who have ever lived is only 1%. Since X is in fact amongst the first 60 billion humans who have ever lived, this means that the total number of humans who will ever be born is more likely to be much closer to 60 billion than to 6,000 billion.
This argument is a strawman, it only works for certain types of distributions (here it assumes humanity has a 50% chance of going extinct after 60 billion humans are born, which is pretty much already assuming what it is claiming to prove). To destroy the argument, all I have to do is choose a different distribution, where humanity has a 99.99% chance of having 6,000 billion humans born before going extinct, and a 0.01% chance of going extinct after 6 billion. Now if I imagine a vast number of different species, all over the universe, that obey this exact same longevity distribution, and I select a member at random from each of those species, and I restrict to the tiny subclass of those random selections in which I got a 59+ billionth member, then I can simply ask-- how many of those civilizations will last to 6,000 billion members? That calculation is easy-- 99.99% of the time I will select from a 6,000 billion member species, and 1/6000 of those I will get someone in the 59+ billionth bin (the bin is a billion people wide), for a grand total of about 1/6000 of the species I sampled. Also, 0.01% of the time I will select from a 60 billion member species, and 1/60 of those will give me someone in the 59+ billionth bin, for a grand total of 1/600,000 of the time. Comparing these frequencies tells us that 99% of the time, my 59+ billionth member, randomly selected from the full population, is part of a 600 billion member species. This refutes the claim of the Wiki article, which holds that we can infer things about the longevity distribution without first assuming anything about it, and that's pretty obviously wrong.
mfb
#25
Nov8-13, 08:59 AM
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Quote Quote by Ken G View Post
The Doomsday Argument is a logical fallacy.
Right.
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.
Ken G
#26
Nov8-13, 11:00 AM
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Quote Quote by mfb View Post
Right.
The first 5% and the last 5% would always get it wrong.
That's another issue, it's harder to say what significance that has. The way it is usually framed is, yes 10% would get it wrong, but there's still only a 10% chance we are among those. I'm saying that if we already know our birth number is 59+ billion, we can not still say that we have a 10% chance of being among those that got it wrong.
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.
Right, we cannot assume correlations don't exist just because we don't know what they are. If we don't use our own birth number, we can treat ourselves as "generic humans", but as soon as we do, we can no longer think we are generic.
goldust
#27
Nov11-13, 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.
Drakkith
#28
Nov11-13, 02:23 PM
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Quote Quote by goldust View Post
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.
Still wouldn't result in the Earth crashing into the Sun. Our orbit would merely become more elliptical. (Or possibly less depending on the circumstances)


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