Indeed, we couldn’t even have hoped for pdfs, or anything based on pdfs, to work any better here. The values of pdfs are just derived from the values of the probability function, and are thus unable to give any more information. And, as I’ve been saying, the fundamental problem with trying to determine possibility/impossibility from the values of the probability function is that there’s just not enough information there; two variables can have the exact same probability distributions (in the sense that, for every set of values, both variables have the same probability of falling in that set), despite one variable being able to take on values that another never can. [For example, the X and Y defined in the second paragraph above have the exact same probability distribution, but X can be 0 while Y can’t and Y can be 3 while X can’t]. It’s perfectly well possible to talk about impossibility/possibility in mathematical terms, but you can’t base it on the values of probabilities alone. You might as well just take it to be a primitive notion in itself, connected slightly to probabilities (in that impossible events must have probability 0), but not determinable from them.
I suppose you may take the point of view that pdfs are defined as the derivative of the cumulative distribution function (rather than, as I was taking them to be, as any function whose integral gives the cumulative distribution function). Or you may have been restricting attention to continuous pdfs. In either of those cases, they are indeed unique (if they exist). Nonetheless, everything else I’ve said still holds.
I’m not sure what you mean by “will admit infinitely many pdfs”.
Based on your definition of Z above, I’d say that Z is uniformly distributed in the set [-1/2, 0) U (0,1/2]
So, f(Z) = 1 for Z in [-1/2, 0) U (0,1/2], and f(Z) = 0, otherwise.
Which means that f(0) = 0.
I disagree that f(0) = 1. As I mentioned above, f(0) = 0, which is consistent with the fact that it is impossible for Z to equal 0.
Note that for Y, I would consider
f(Y) = 1 when (Y in [-1,0) U (0,1]) or (Y = 3), and f(Y) = 0 otherwise.
So, f(3) = 1, which means that Y = 3 is possible, though it has a probability of zero, which is consistent with what I said in the previous post.
Overall, I agree that if we get into the details it can get hairy. To make things simpler, I should have noted that my example in the previous post was restricted to pdfs that are continuous functions. Otherwise, we’ll need to talk about pmf’s, Dirac Delta functions, etc (and, from what I know, mathematicians don’t consider Dirac Delta functions as “real functions”)
I was just trying to set up an example where impossibility was denoted by a mathematical concept (i.e. the value of the pdf at a certain point).
As an aside, it seems that cdfs behave much “nicer” that pdfs, and should be used in the more rigorous proofs. (e.g. when you have a continous density with a mass at some point, the pdf would be something like a continous function plus a Dirac Delta at the point of the probability mass. From what I know, this makes some purists nervous, and won’t consider such a pdf a ‘function’. However, the cdf is much nicer, having only a step where the probability mass is, so it behaves much more like a function)
Yeah, I’ve now clarified my remarks on this.
Correct.
Why would f(0) = 0? You told us:
. Keeping that in mind, we should take f(0) to be 1, as Pr(Z in [0 - D/2, 0 + D/2]) = D * 1.
It’s not consistent with your “probability of being close” criterion for the values of the pdf.
Again, though, not consistent with your “probability of being close” criterion for the values of the pdf. To be consistent with that, you need f(3) = 0.
Hairy indeed. But note that both the pdfs I proposed are continuous, while both the ones you propose in this post are discontinuous at the points of interest.
It was a good effort, but I’d say it failed to solve the problem of determining possibility from probability, because that problem can’t be solved. Possibility/impossibility is simply not determinable from probabilities, for all the reasons I’ve outlined. Again, consider the case of the X and Y which have the same probability distributions but different possibilities/impossibilities. Clearly, there isn’t enough information in the probability distributions from which to recover all the possibility/impossibility information. But as I’ve said before, this doesn’t mean possibility is not a mathematical concept; just that it’s not a purely probabilistic concept, as such.
Yeah, I agree, cdfs are usually “nicer” to work with, or feel “cleaner” or vaguely aesthetically preferable in some way, even though I’m not terribly opposed to working with generalized functions like the Dirac delta in this context.
While it doesn’t undermine the point you were making, I feel compelled to note that the number of distinct bridge hands is much, much smaller than that: C(52,13) = 52!/(13!39!) = 635,013,559,600 to be precise.
how many Indistinguishables are there? How can we distinguish them?
Yes, that’s true for surfaces in 3-dimensional Euclidean geometry. However, it’s not true of surfaces in the real world, because of the finite size of atoms and sub-atomic particles, and because of quantum effects. Reality is only an approximation of Euclidean geometry: a theoretical sphere has an infinite number of points on its surface, but a real sphere (or even an approximate sphere (like a football or the Earth) has only a finite number of points on its surface.
I don’t get the problem in the OP.
If you have an infinite number of coins, this is a very different situation than the one in which you are talking about the probability of one particular coin. The latter statement is poorly phrased: it is a statistical statement about probability approaching 50% as more and more coin flips are made. That has no bearing on the fact that you’ve previously stipulated that there are infinite number of coins and thus are looking at the few that never flip heads. The vanishingly improbable occurance is being DEFINED as being inevitable with an infinite number of coins, and coins approaching a perfect 50% is an aggregate statistical conclusion, not a fact for what all coins must do over a very very long finite period or even an infinite one.
You only get the larger number, 3.95424*10^21 if you take the order you receive the cards as being significant. In this model, A-K would be different than K-A. From a bridge point of view it’s invalid, but looking at the probability of being dealt a specific hand it’s not.
Yer absolutely right Rufus, sorry about that. I had a brain fart and calculated the cumulative probability of getting each card given that the previous ones had already been dealt, which isn’t the same thing. Thanks for the correction.
So okay, if you were randomly dealing one bridge hand per second from the beginning of the universe to the present moment, you would almost certainly get any possible hand at some point. But not if you were dealing at the same rate for, say, a hundred years. Which in practical terms still makes it extremely improbable that any particular bridge hand should ever get dealt in the course of ordinary play—and yet, any particular hand that you encounter obviously has been dealt. So my analogy, although less gee-whizzical numerically than it originally appeared, basically still stands.
Why do you equate points to sub-atomic particles?
It shouldn’t be sub-atomic particles, it should be Planck length. In “the real world” anything smaller than Planck length is meaningless.
I don’t think that is exactly true. It is not like the universe is a three dimensional grid where everything moves one plank length at a time.
Leaving aside coins, the premise of the OP seems to be that God exists because the universe is too improbable to exist unless God intentionally created it. But the flaw with that argument is that it presumes its own answer. Otherwise you have to address the issue of the probability of God’s existence and which is more improbable: a self-created universe or a universal creator?
My read of that cite is that the Planck length is simply a limit on measurement, not on distance. After all, we’re not proposing that any particles are that close together, just that there’s infinite variation on how the object could have landed.
It’s not as if the sphere lands on a neat tripod of the three lowest atoms, after all. It’s landing position is determined by the interaction of the various energy fields and forces at that level - which are continuous, I believe. So it could have landed a little less than a Planck length to the left of where it did, balanced on those same fields in a slightly different position.
Leaving aside coins, the OP presumes the creation of the universe, life, sentience, etc to be infinitely improbable, without any actual reason to. I mean, sure, it doesn’t seem to happen every day on every street corner, but that doesn’t make it impossible or anything. Just rare. (At least around here.)
This is a thought discussion about a strawman, essentially.
Yeah, the God connection is not particularly well-conceived. I say we all just drop it and keep the discussion focused on the issues of probability in themselves.
Absolutely. If I got dealt a nondescript hand like the following:
S: J 7 5 4
H: K 6
D: Q 10 3
C: Q J 9 7
As you say, I could exclaim, “look at this hand I just got dealt - did you know that the odds against getting this hand were 635 billion to 1 against?!” and I’d be absolutely correct. Everything that happens is extremely improbable at some level, but (with the possible exception of the Universe’s big-banging into existence in the first place) something’s gotta happen. So just that it’s extremely, extremely improbable for X to happen, doesn’t mean that a Designer had to be behind it.
But could a thousand monkeys with an infinite number of coins in a slower growing infinite amount of time flip a coin an infinite number of times while typing hamlet an infinite number of times on three typewriters while humming the national anthem of Kazakhastan?