ultrafilter
Well, almost. I should have specified that the range of physical states is finite (which is, of course, the case for our current “best guess” cosmology).
Even then, you have a point that any particular state need not recur. It is possible that a pattern of repetition will emerge which bypasses some states. However, it is guaranteed that at least some states recur infinitely many times.
It’s hard to explain the difference, because there is none. In the infinite case, the probability is 0, just like it is for any particular sequence of flips, but it can still happen.
Actually, ultrafilter, I believe his point was that one cannot flip a coin infinitely many times. Infinity is not a number.
I may be wrong, but that is how I interpreted his “difference”.
no, there is definitely a difference between “infinity” and “any finite number”.
if you do something an infinite number of times, and there is a probability greater than 0 of it achieving a particular state for each trial, it is guaranteed to occur.
the only way it wouldn’t is if the probability was 0, for instance if the coin was weighted in such a way that landing on heads was impossible.
Ramanujan
If that is what you meant, then I must concur with ultrafilter. The probability of any discreet event (such as an unending series of “tails”) given an infinite probability field is 0. Thus, there is never a guarantee that you will achieve a “heads”, though you flip a fair coin until the Universe ends.
Another way to look at this is to realize that each flip is an independent event. It matter not how many tails have come before, the next flip has a “fair” chance to come up tails again. This is one of the common (usually implicitly felt rather than reasoned) fallacies which turns gamblers into paupers.
Nevertheless, I do agree that there is a difference between flipping a coin countably many times and flipping it infinitely many times. One can be acheived, the other cannot.
The way I learned it was that the difference is that a finite number of flips (where the number of flips approaches infinity) will have a probability of heads coming up that approaches 1, whereas the probability of heads coming up for an infinite number of flips is exactly 1. (Or, the probability of all tails coming up approaches 0 in the first case, but is exactly 0 in the second case.)
You’re mistaken here. The probability is zero that it doesn’t occur, but that doesn’t mean that there’s a guarantee. And there isn’t.
i agree with loinburger. they are independent events, but you have a greater chance of flipping more heads the more you flip the coin. so the probability of getting “at least one heads” is greater if you flip a coin 20 times than say 1 time. so if you flip it 999 byzantillion times you might not get a heads, but if you keep flipping forever, you are guaranteed to get at least one.
the point is that there is a difference between arbitrarily many and infinitely many. you can keep flipping until you get a heads, for example. as long as there is a chance it may come up on a given flip, it will come up eventually.
also note that this says nothing about the probability of tails coming up. this isn’t “on a given flip, the probability of heads coming up is 1.” this is “after infinitely many flips the probability of heads coming up at least once is 1.” they are different probabilities.
Last I checked, a probability of zero means that the event will not happen, by definition. Similarly, a probability of 1 means that, by definition that event will occur. Granted, these are definitions I learned in basic statistics class, so maybe when you get to Really Super Hard Uber-Statistics 101 they change things around on you, but unless that’s the case, it seems absurd to say, “Just because something has a probability of 0 doesn’t mean it can’t happen.”
The odds of me rolling a standard 6-sided die and getting a number between 1-6 is 1; does that mean that there a chance that I could roll that die and get “34”?
Jeff
Oh goody, the old “infinite coin flips” debate.
ElJeffe, your last post demonstrates the difference between experiments with a finite number of outcomes and experiments with an infinite number of outcomes. In the latter case, it’s quite possible for a particular event to have probability zero but still happen.
In fact, in the case of flipping a coin an infinite number of times, every outcome has probability zero. Yet one of the outcomes has to happen anyway. And it’s certainly possible that the outcome could be “TTTTT…” (i.e. all tails); in fact that outcome is just as likely as any individual outcome of the form “TTHTTHHTHTTHHT…”.
“an infinite number of flips” is simply sloppy language. There ain’t no such beast. So, the probability of heads coming up in an infinite numbe of flips is not 1, it is undefined.
By convention, one might use “the probability of an infinite number of flips . . .” to indicate a limit as the numbe rof flips approach infinity, but I don’t like such conventions since they create exactly these sorts of confusions.
Infinity is not a number. “An infinite number of” is fine as a linguistic convention, but it is bad math.
Consider it as a sequence of binary digits, one for each natural number. In that case, there’s no limit. We really are dealing with a countable number of them.
Please provide a proof of this statement.
And the above post, of course, is bad tpying.
I have no idea what you mean, here. The ability to establish a mapping to the natual numbers does not mean that “infinity” is a number. It isn’t.
One can flip coins forever, assuming that one lives forever, but one will never have flipped an infinite number of coins. I agree with you on how the probability field should be evaluated; there is no guarantee that a “heads” will ever appear. That does not mean that there is no difference betwen “any finite number of times” and “an infinite number of times.”
“infinity” is not a number. But how many elements are there in the set {0, 1, 2, …}? That’s how many coin flips we’re talking about here.
The thing is to not get too caught up in the physical process of flipping the coin. The result is what matters. All we’re doing is taking each natural number and assigning to it either a 0 or a 1 (or an H or a T, if you like). There’s nothing ill-defined about that, and it is a particular number of coin flips, if not a finite one.
Nope.It is not a “particular number of coin flips” (or anything else).
How many elements are contained in an infinite set? Colloquially, an infinite number. Precisely, not a number at all.
You need to read up on transfinite arithmetic. As a mathematician, I can tell you that it most certainly is a number–just not one you’re used to dealing with.
I’m somewhat familiar (though it has been a while) with Cantor. I certainly haven’t kept up with new developments in the field.
While it is certainly the case that Cantor termed his cardinalities transfinite numbers, I think that most (though not all, obviously) mathematicians avoid referencing cardinalities as numbers, since transinfinites lack so many poperties that one normally associates with numbers: unity, reciprocals, zero element, etc. Personally, I have always preferred to conceptualize transfinites as magnitudes, but that’s just me.
We can argue, I suppose, over whose usage is most consistent with the broad spectrum of mathematicians today, but how about if I simply rephrase it:
Infinity is not a number in the sense which is commonly understood and reasonable to insert into a common arithmetic expression.
Most mathematicians just consider them numbers. They do have different properties from the more familiar numbers. But then again, there is a characterization of the naturals with no 0, integers don’t have reciprocals, and the complex numbers have no ordering.
**
Infinity is not. Aleph0, the cardinality of the integers, is, although you have to treat it rigorously.