What are the chances of my flipping a dime an infinite number of times and having it land on heads every time?
Small.
Zero. The chance that you’ll get all heads if you flip the dime n times is 2[sup]-n[/sup]. If you get infinitely many heads in a row, you have to have gotten heads for the first n tosses and then some, so the probability has to be no larger than 2[sup]-n[/sup] for every n. Probabilities can’t be negative, so zero is the only possibility.
Well, it’s a poorly constructed question. You can’t flip a coin an infinite number of times. In a way, the odds of flipping a coin AT ALL an infinite number of times is 0, let alone getting heads each time.
As the number of coin flips approaches infinity, the probability that all the flips are heads (as calculated before any flips take place) approaches zero.
Correct. You don’t even need the whole question though. The answer to “What are the chances of my flipping a dime an infinite number of times?” is also zero.
Wouldn’t it be more accurate to say that as the number of flips approaches infinity, the probability approaches zero?
ETA: Or what Chipacabra said.
Insert your favorite disclaimer about the relationship between mathematical models and reality here. No, you can’t flip a coin infinitely many times, but the model that you get if you assume you can is useful, and given that model, the right answer is zero.
It depends on the dime. If it is a normal dime, P=0. If it is a two-headed dime, P=1.
Also, who flips a dime, anyway? They’re too small.
Any repeating pattern (including all heads) has to be zero, and going further any combo of a infinite series of flips, if we allow it, is also zero, so every infinite # of coin flips is unique, never again to be duplicated and therefore is impossible thereby zero.
Is it a fair coin?
If you flip it an infinite number of times and it always comes up heads, then it isn’t a fair coin.
Yes. This is an example of a “limit”. The probability of getting all heads decreases asymptotically as the number of trials goes to infinity.
kanicbird writes:
> Any repeating pattern (including all heads) has to be zero, and going further
> any combo of a infinite series of flips, if we allow it, is also zero, so every infinite
> # of coin flips is unique, never again to be duplicated and therefore is
> impossible thereby zero.
I can’t tell what you’re saying here. If it’s what I think you’re saying, it’s wrong. That’s like saying that if one picks a random real number, the chance that any particular real number is chosen is 0, so the sum of the chances for all real numbers is a sum of 0’s, so the chance of any real number at all being chosen is 0. You can’t do that kind of summing for infinite sums. The chance of any given real number being chosen is 0, but the chance of some real number being chosen is 1. Similarly, the chance of any given infinite pattern of flips is 0 (since it’s equivalent to the binary representation of a number between 0 and 1), but the chance of some pattern from all possible infinite patterns of flips is 1.
I wouldn’t necessarily want to say this. Why do you say this? Is it simply because a “fair coin” (which I would take to mean something modelled mathematically as producing a number of independent events each equally probable to be heads or tails), when flipped infinitely, has probability zero of coming up heads every time?
For any particular sequence, the probability that that sequence will be produced if you flip a coin an infinite number of times is zero, by the line of reasoning given by ultrafilter. Still, if you flip the coin infinitely often, some sequence comes up. So we draw a distinction between “Happens with probability zero” and “Never happens” (a distinction I know you’re aware of, but which may as well be brought up now).
That having been said, I’m not completely opposed to trying to take a position something like “A fair coin will never produce a sequence with any priorly-describable/existent property with probability zero”, so that flipping a fair coin infinitely often, in some sense, must make a fresh infinite sequence of bits that didn’t previously exist and adds it to the universe. I’ve left the technical details out, but in fact, interestingly enough, you can think of this as precisely what is going on in Cohen’s proof that ZFC can’t consistently disprove the continuum hypothesis (as shown most clearly in the measure-theoretic presentation of this result given by Dana Scott).
ETA: So, there is something to the line of thought being proposed by kanicbird, and indeed, that’s what underlies the proof: if you flip K many coins infinitely many times, you produce K many necessarily distinct series of bits, in some sense, showing that the number of series of bits afterwards must be at least what K was, in some sense. Now, the technical details of all this (passing between an “old universe of sets” and the “new universe of sets” and so on) are, of course, presumably nothing kanicbird was aware of at any formal level, but he is getting at a very useful and productive intuition. One must note the contradictions that lurk around it, as Wendell did, but it’s not completely off-base to have such thoughts either.
So, in summary: I wouldn’t necessarily want to say that a “fair coin” tossed infinitely often can’t come up heads every time, since there are tricky paradoxes to be addressed with such thinking, but I might sometimes want to say it, since such thinking can still be to some extent tamed into consistency and have intuitive and fruitful consequences… So, uh. Depending on what I’m doing, boo or yay.
(Hm, with such rampant ambivalence, why’d I bother making a post?)
(To clarify, the useful intuition that can be extracted from what kanicbird was saying is that every infinite coin-toss must produce a new series of bits distinct from all previous infinite coin-tosses; the concluding “and therefore is impossible thereby zero” is something I don’t know how to parse, but presumably the result of then foundering upon the paradoxes that bedevil such lines of thought, the avoidance of which requires some more sophisticated mathematical/logical care)
The probability that a fair coin tossed infinitely often comes up tails at least once is one. In fact, a fair coin will come up heads infinitely often and tails infinitely often with probability one.
Yes, I know. Isn’t that part of what I said? My discussion was about to what extent we should or should not identify “probability 1” with “guaranteed”. Is a fair coin guaranteed to come up tails at least once? That’s the statement I am exploring my ambivalence towards (of course, all there is to the ambivalence is pointing out that both “Yes” and “No” correspond to useful mathematical models of infinite coin flipping to think about, each with their own unintuitive and intuitive aspects).
Sorry, overnegated. Meant to say “proof that ZFC can’t consistently prove the continuum hypothesis”.
By “fair coin”, I mean that each flip independently has a probability of 1/2 of coming up heads, and a 1/2 probability of coming up tails. But what does probability mean? When I say that a coin has a 1/2 chance of coming up heads, I mean that if I flip it a number of times, the ratio of the number of heads to the number of flips approaches 1/2 in the limit as the number of flips approaches infinity. If I flip it an infinite number of times and get heads every time, then the limit of that ratio isn’t 1/2, it’s 1, and so I say that the coin is not fair.