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extropalopakettle
No offense, but....
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 Posted: Wed Sep 07, 2011 4:42 pm Post subject: 161 |
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| BraveHat wrote: |
| Thok wrote: |
| You've missed 3: reject the idea that we can directly add up an uncountable number of numbers in a reasonable way. Which makes sense, since the most we ever add together is a countable number (in infinite series). |
That might work for my specific example, but I could easily make the same case for a countable infinite set. To wit: if an abstract dart is thrown at the natural number line with uniform probability, then the probability of 5 being hit being exactly 0 contradicts that fact that 5 has a chance to be hit. As extro points, countable versus uncountable doesn't really matter to the particular type of contradiction. |
I should have took that back when I crossed out the question that followed it.
If an abstract dart is thrown at the natural number line with uniform probability, and the probability of each natural number being hit is exactly 0, then the sum of the probabilities is 0, which contradicts the definition of probability (probability of all possible outcomes must add up to 1).
And of course if the probabilities for all numbers are greater than 0, their sum is also not 1 (it's undefined, infinite).
For the case of reals, the continuous uniform probability distribution assigns a probability of r2 minus r1 that the chosen real is in [r1, r2]. The probability that the number chosen equals r is simply the probability that it's in [r,r], i.e. 0. |
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Zag
Tired of his old title
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| Posted: Wed Sep 07, 2011 4:49 pm Post subject: 162 |
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You continue to refuse the concepts of infinity.
I won't even discuss throwing your abstract dart at the entire number line, but I'm willing to discuss throwing it at the real number line in the range [0,1] with uniform probability. What are the chances of hitting precisely 0.5? (By "precisely 0.5" I mean 0.50000...) Let me ask, instead, what are the chances of NOT hitting precisely 0.5
One way to simulate this operation is to say that we will generate the number through an infinite process of choosing digits. We start with 0. and continue to choose digits randomly.
For the first digit, there's only 1/10 chance of choosing a 5, so the chances of not hitting precisely 0.5 are, so far, 0.9.
Assuming we do choose a five, then we would need to choose a zero for the next number. There's again only 1/10 chance, so the chance of failing to stay on target for our precise 0.5 is now
0.9 + (0.1) * (0.9) = 0.99
If we keep up this process to infinity, we fine that the chance of choosing something other than precisely 0.5 is
sum
(n=1 to infinity)
(0.9 * (0.1) n ) = 0.9999...
Since we already established that Achilles DOES catch the tortoise, we know that this number is EXACTLY equal to 1. Not very, very, very close, but exactly equal.
Intuitively, you can see that this is also the case. If you are generating the random numbers and adding digits, you stop as soon as you generate a number other than 0 (after that initial 5), but you have to keep going if you generate a 0. Your only termination is to fail to produce the precise number. |
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Zag
Tired of his old title
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| Posted: Wed Sep 07, 2011 4:52 pm Post subject: 163 |
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| I suppose we could use the same process for choosing a positive integer uniformly across the entire number line. We generate the digits in reverse, starting at the 1's position. To actually hit any number you name, we have to generate it, then generate zeroes forever. |
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raekuul
Lives under a bridge & tells stories.
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| Posted: Wed Sep 07, 2011 8:11 pm Post subject: 164 |
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| Which we do without even thinking about it most of the time. |
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BraveHat
Last of the Daedalians
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| Posted: Wed Sep 07, 2011 8:36 pm Post subject: 165 |
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| Zag wrote: |
| If we keep up this process to infinity, we find that the chance of choosing something other than precisely 0.5 is...EXACTLY equal to 1. Not very, very, very close, but exactly equal. |
But the question is, in layman's terms, does that probability mean that choosing something other than precisely .5 is "almost surely" or "absolutely surely"? If it's "almost surely", then "probability 1" has two different meanings when talking about the infinite, and those meanings should be distinguished in notation. If it's "absolutely surely" then it contradicts the plain fact that "precisely .5" can be picked.
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BraveHat
Last of the Daedalians
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| Posted: Wed Sep 07, 2011 8:38 pm Post subject: 166 |
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...and no "Airplane!" jokes  (RIP Leslie Nielsen)
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"I am declaring it a terrible tragedy for me to die. You may disagree..." --Antrax |
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BraveHat
Last of the Daedalians
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| Posted: Wed Sep 07, 2011 8:52 pm Post subject: 167 |
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| Zag wrote: |
| Your only termination is to fail to produce the precise number. |
And it's possible it could never terminate, and just keep producing zeroes forever
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"I am declaring it a terrible tragedy for me to die. You may disagree..." --Antrax |
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BraveHat
Last of the Daedalians
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| Posted: Wed Sep 07, 2011 9:11 pm Post subject: 168 |
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The abstract dart has a tip dimension of 1 point, so it is going to hit an exact point on that infinite number line, no matter what kind of line it is. If we consider x to be any digit (including 0), then that exact point will have a value of ...xxx.xxx... where the number so written goes off to infinity both on it's left side and right side. (even if the number is "precisely .5" the infinite left x's would be all 0s and the infinite x's to the right of the 5 would be all 0's). So, because it will hit that precise number, that precise number cannot have an "absolute never" chance of being hit.
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"I am declaring it a terrible tragedy for me to die. You may disagree..." --Antrax |
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BraveHat
Last of the Daedalians
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| Posted: Wed Sep 07, 2011 9:18 pm Post subject: 169 |
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If the typical precise point on the real number line could be written, the middle of it would look something like this:
...1740183470139247810293478102347.710438710934701384091734099...
if we were looking at a negative number, I suppose the hyphen would have to be written on the top or bottom since the left goes out to infinity.
_________________
"I am declaring it a terrible tragedy for me to die. You may disagree..." --Antrax |
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Thok
Oh, foe, the cursed teeth!
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| Posted: Wed Sep 07, 2011 9:27 pm Post subject: 170 |
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| BraveHat wrote: |
| To wit: if an abstract dart is thrown at the natural number line with uniform probability |
We've already explained why you can't throw a dart at the natural number line with uniform probability, or more generally why you can't have a uniform probability on any countably infinite set.
(There's also no notion of uniform probability on the real line, but that's a separate argument.)
| Quote: |
| The seeming contradiction only points to the failure of mathematicians to notate the difference. |
On finite sets, "almost never" and "absolutely never" mean the same thing, for the correct definition of "almost never". On infinite sets, absolutely never doesn't generalize to any useful notion of probability, while "almost never" generalizes to an incredibly useful notion that has obvious real world applications.. Unless you don't like the normal curve, in which case you've just rendered all of statistics and polling useless. |
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BraveHat
Last of the Daedalians
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| Posted: Wed Sep 07, 2011 9:46 pm Post subject: 171 |
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| Thok wrote: |
| On infinite sets, absolutely never doesn't generalize to any useful notion of probability, while "almost never" generalizes to an incredibly useful notion that has obvious real world applications.. Unless you don't like the normal curve, in which case you've just rendered all of statistics and polling useless. |
Hmm...could you give me a real world example of how probability in infinite sets is used in statistics?
(the uselessness of statistics and polling is another debate )
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"I am declaring it a terrible tragedy for me to die. You may disagree..." --Antrax |
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Thok
Oh, foe, the cursed teeth!
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| Posted: Wed Sep 07, 2011 9:49 pm Post subject: 172 |
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| Zag wrote: |
| I suppose we could use the same process for choosing a positive integer uniformly across the entire number line. We generate the digits in reverse, starting at the 1's position. To actually hit any number you name, we have to generate it, then generate zeroes forever. |
The probability of generating zeroes forever is 0 (well the limit as n goes to infinity of n*(9/10)^n, which is the same thing.) Which is why we don't make integers this way.*
(*Unless you are working with the p-adic integers. But that's beyond the scope of this thread and then the fractional part has finitely many digits.) |
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Thok
Oh, foe, the cursed teeth!
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| Posted: Wed Sep 07, 2011 9:52 pm Post subject: 173 |
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| BraveHat wrote: |
| Hmm...could you give me a real world example of how probability in infinite sets is used in statistics? |
The normal curve, Student's t curve, the Chi squared distribution, the gamma distribution, ....
If you don't know how the normal curve relates to probability on an infinite set, read the wikipedia article. Then reread it. |
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extropalopakettle
No offense, but....
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| Posted: Thu Sep 08, 2011 12:36 am Post subject: 174 |
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| Thok wrote: |
| BraveHat wrote: |
| Hmm...could you give me a real world example of how probability in infinite sets is used in statistics? |
The normal curve, Student's t curve, the Chi squared distribution, the gamma distribution, ....
If you don't know how the normal curve relates to probability on an infinite set, read the wikipedia article. Then reread it. |
Simple example: Flip a coin until it comes up heads. Count how many flips it takes. There are an infinite number of possible outcomes, each with their own non-zero probability. Is that "real world" enough? |
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BraveHat
Last of the Daedalians
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| Posted: Sat Sep 10, 2011 12:23 am Post subject: 175 |
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| extro wrote: |
| Simple example: Flip a coin until it comes up heads. Count how many flips it takes. There are an infinite number of possible outcomes, each with their own non-zero probability. Is that "real world" enough? |
I'm not sure that's what I'm looking for, because I'm seeking an example of an infinite set where "almost never" applies, which is a uniform probability, yes?
| wikipedia on 'almost never' wrote: |
If an event is almost sure, then outcomes not in this event are theoretically possible; however, the probability of such an outcome occurring is smaller than any fixed positive probability, and therefore must be 0.
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Even though each possible outcome of your real world situation has a smaller probability the more flips are tossed, none of the real world outcomes have a probability which is smaller than any fixed positive probability (unless you say infinite tosses are a possible real world outcome, but that hasn't been shown to be possible in the same sense. It's only possible that it's possible) So, rather, to find a real world use of "almost never", I would need a real world use of infinite sets with uniform distribution. Unfortunately, it's taking me a while because I never took statistics and Thok's wikipedia article is a lot to take in.
_________________
"I am declaring it a terrible tragedy for me to die. You may disagree..." --Antrax |
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Thok
Oh, foe, the cursed teeth!
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| Posted: Sat Sep 10, 2011 1:50 am Post subject: 176 |
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| BraveHat wrote: |
| I'm not sure that's what I'm looking for, because I'm seeking an example of an infinite set where "almost never" applies, which is a uniform probability, yes? |
No. You merely want a continuous distribution on an infinite set. The probability of getting any exact point on a normal distribution is 0, but it's not uniform in anyway (the interval [-1,1] in much more probable than the interval [100,102]).
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The main reason why the normal curve is important is the central limit theorem: if you take any probability distribution that has a mean and a standard deviation and look at the probability distribution you get when you look at the average of a lot of independent experiments, it approaches the appropriate normal distribution as the number of experiments gets large.
For example, consider flipping a million coins and asking what's the odds that you get between .4 and .6 heads on average. (Here the probability distribution we start with is a coin flip, and we're looking at what happens when we do that experiment a lot.) While there is an exact answer, it requires adding up a lot of binomial coefficients and therefore is mostly hopeless to do (and runs into the problem of rounding error.)
Instead, you can use a normal approximation to the flip a million coins distribution, and you get an answer that's much easier to compute using numerical integration.
(This is also why polling works: since you take the average opinion of a lot of people, you don't have to know what the original distribution of opinions was, since the distribution of the average opinion will look normal.) |
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extropalopakettle
No offense, but....
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| Posted: Sat Sep 10, 2011 4:20 am Post subject: 177 |
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| Thok wrote: |
| BraveHat wrote: |
| I'm not sure that's what I'm looking for, because I'm seeking an example of an infinite set where "almost never" applies, which is a uniform probability, yes? |
No. You merely want a continuous distribution on an infinite set. The probability of getting any exact point on a normal distribution is 0, ... |
I think that fails as a "real world" example though, as there's no real world way of selecting a point from a continuum. A real world dart thrown at a real world line, measured with real world methods, yields something from a countable, and perhaps finite, set.
In the real world, are there possible events with probability 0? |
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BraveHat
Last of the Daedalians
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| Posted: Sat Sep 10, 2011 4:34 am Post subject: 178 |
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| Thok wrote: |
| No. You merely want a continuous distribution on an infinite set. |
Oh, I think I made a mistake. Extro's infinite set was countable, yes? His real life situation was flipping a coin until it comes up heads, and his infinite set was the set of all possible outcomes {H, TH, TTH, TTTH,...} That's why "almost never" doesn't apply to elements of that set, because it's countable (H has a .5 chance, TH has a .25 chance, TTH a .125 chance, and TTTH a .625 chance, etc) That's why his example doesn't meet what I'm looking for.
| Thok wrote: |
| The probability of getting any exact point on a normal distribution is 0, but it's not uniform in anyway (the interval [-1,1] in much more probable than the interval [100,102]). |
But is there a way to notate what probability 0 is greater than anther probability 0?
_________________
"I am declaring it a terrible tragedy for me to die. You may disagree..." --Antrax |
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Thok*
Guest
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| Posted: Sat Sep 10, 2011 10:05 am Post subject: 179 |
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| BraveHat wrote: |
But is there a way to notate what probability 0 is greater than anther probability 0? |
Yes. Instead of keeping track of the probability at a point, you keep track of the rate of change of probability at a point.
The probability of an interval in the the normal distribution is the area under the appropriate segment of the normal curve. So while any point has zero probability (the area under a point is zero), passing by points near the center of the normal curve increases the area faster than passing by points far from the center. |
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BraveHat
Last of the Daedalians
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| Posted: Sat Sep 10, 2011 11:00 pm Post subject: 180 |
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| extro wrote: |
| In the real world, are there possible events with probability 0? |
This one is tricky semantically. I mean, theoretically, couldn't you take any single possible real world event and look at it as part of an uncountable infinite set, thus giving it a probability of 0? For example it's possible that it could rain tomorrow. "Raining tomorrow" is one element of the set of all possible future events, which is uncountable. So in that sense, we could say there is a probability of 0 that it will rain tomorrow. In that case, I would like to consider my having to go to work on Monday a member of that same set, so that I can say there is a 0 probability of me having to go to work on Monday 
_________________
"I am declaring it a terrible tragedy for me to die. You may disagree..." --Antrax |
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BraveHat
Last of the Daedalians
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| Posted: Sat Sep 10, 2011 11:05 pm Post subject: 181 |
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Thok, extro, Zag, raekuul, Nsof and others, thank you all for taking the time to answer these questions and explaining your answers the best of your abilities. I know I can seem ridiculous a lot of times with my questions, and I'm sorry if sometimes I start acting too much like I know what I'm talking about, but I really am enjoying this conversation!
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"I am declaring it a terrible tragedy for me to die. You may disagree..." --Antrax |
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extropalopakettle
No offense, but....
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| Posted: Sun Sep 11, 2011 3:35 am Post subject: 182 |
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| BraveHat wrote: |
| ... couldn't you take any single possible real world event and look at it as part of an uncountable infinite set, thus giving it a probability of 0? For example it's possible that it could rain tomorrow. "Raining tomorrow" is one element of the set of all possible future events, which is uncountable. |
Infinite, but not uncountable.
Take the simple example of the dart thrown at a number line. Any measurement of its position will have a finite representation (thus part of a countable set of possibilities) and a degree of precision, thus being a specification of some interval in which the dart landed, rather than an exact point, and thus having an associated non-zero probability. |
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BraveHat
Last of the Daedalians
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| Posted: Sun Sep 11, 2011 4:04 am Post subject: 183 |
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| extro wrote: |
| I wrote: |
| ... couldn't you take any single possible real world event and look at it as part of an uncountable infinite set, thus giving it a probability of 0? For example it's possible that it could rain tomorrow. "Raining tomorrow" is one element of the set of all possible future events, which is uncountable. |
Infinite, but not uncountable.
Take the simple example of the dart thrown at a number line. Any measurement of its position will have a finite representation (thus part of a countable set of possibilities) and a degree of precision, thus being a specification of some interval in which the dart landed, rather than an exact point, and thus having an associated non-zero probability.
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Yes, but my point was that the set of all possible events is uncountable and since any possible event is a member of that uncountable set, it can be considered to have a probability of 0 with respect to that set. Or is that uncountable set one in which probability is ill-defined to begin with?
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"I am declaring it a terrible tragedy for me to die. You may disagree..." --Antrax |
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extropalopakettle
No offense, but....
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| Posted: Sun Sep 11, 2011 5:42 am Post subject: 184 |
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| BraveHat wrote: |
| Yes, but my point was that the set of all possible events is uncountable ... |
Why is that so? I'll accept infinite, but why uncountable? |
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Thok*
Guest
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| Posted: Sun Sep 11, 2011 12:30 pm Post subject: 185 |
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| extropalopakettle wrote: |
| Why is that so? I'll accept infinite, but why uncountable? |
It's a representation of an interval [0,1], and thus uncountable.
As I said before, the right solution is to accept that addition works well when combining a finite amount of summands, sort of works OK when combining a countable number of summands, and horribly breaks down when combining uncountably many summands. Which is why we switch to defining probability in terms of integration instead.
And then the point is that probability defined that way works as we expect it on intervals, and there are theorems that allows us to consider the probability distributions defined by integrals as infinite limits of the probability distributions defined on finite sets. |
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extropalopakettle
No offense, but....
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| Posted: Sun Sep 11, 2011 3:47 pm Post subject: 186 |
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| Thok* wrote: |
| extropalopakettle wrote: |
| Why is that so? I'll accept infinite, but why uncountable? |
It's a representation of an interval [0,1], and thus uncountable. |
Bravehat suggested that in the real world, there are events that occur that are from an uncountable set of possible outcomes, and thus that possible real world events can have probability 0. Is that actually the case, is what I'm asking. |
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Thok*
Guest
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| Posted: Sun Sep 11, 2011 3:54 pm Post subject: 187 |
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| extropalopakettle wrote: |
| Bravehat suggested that in the real world, there are events that occur that are from an uncountable set of possible outcomes, and thus that possible real world events can have probability 0. Is that actually the case, is what I'm asking. |
It's a good approximation of the case.
I mean, in some sense we're asking how things work in the universe at a very small scale, at which point quantum mechanics starts to laugh at everybody. But yes, despite actually happen, an exact event has probability 0. We mostly measure ranges of events. |
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extropalopakettle
No offense, but....
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| Posted: Sun Sep 11, 2011 4:00 pm Post subject: 188 |
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| Thok* wrote: |
| extropalopakettle wrote: |
| Bravehat suggested that in the real world, there are events that occur that are from an uncountable set of possible outcomes, and thus that possible real world events can have probability 0. Is that actually the case, is what I'm asking. |
It's a good approximation of the case. |
And a 0 probability is a good approximation to a very small non-zero probability.
| Thok* wrote: |
| I mean, in some sense we're asking how things work in the universe at a very small scale, at which point quantum mechanics starts to laugh at everybody. But yes, despite actually happen, an exact event has probability 0. We mostly measure ranges of events. |
Not sure what you're saying here. "an exact event has probability 0" ... in the real world?
That we mostly measure ranges of events does not argue for the actual events being uncountable. |
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Nsof
Daedalian Member
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| Posted: Sun Sep 11, 2011 8:20 pm Post subject: 189 |
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| Thok* wrote: |
| It's a representation of an interval [0,1], and thus uncountable |
Why?
I am no expert at physics but if there is
a) a minimal quanta of time,space
b) a finite amount of mass
doesn't that make anything physical at most countable and maybe finite? (if so then obviously some events will have greater than 0 probability of occurring)
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Will sell this place for beer |
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extropalopakettle
No offense, but....
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| Posted: Sun Sep 11, 2011 11:50 pm Post subject: 190 |
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Hmmm .... this is getting interesting.
If there's an infinite number of particles of matter, and infinite quantized space, and we just consider which locations of space are occupied by matter, then I think we have an uncountable number of possible states.
With a finite number of particles of matter, it's countable.
(gross simplification of course) |
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Thok*
Guest
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| Posted: Mon Sep 12, 2011 12:04 am Post subject: 191 |
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| Nsof wrote: |
| a) a minimal quanta of time,space |
This is what I basically meant by "quantum mechanics laughs at us". The problem is that a particle, even when quantized, isn't discrete: it's a probability distribution with a location that only gets realized when we observe it. And quantum mechanics doesn't force abstract particles to live on some fixed grid.
In any case, it's either finite but extremely large or uncountable. |
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Nsof
Daedalian Member
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| Posted: Mon Sep 12, 2011 9:54 pm Post subject: 192 |
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If space is continuous then even a finite universe would be uncountable. No?
I earlier said that irrationals, as an abstract concept, cannot correlate with physical quantities. that was obviously wrong.
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extropalopakettle
No offense, but....
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