If a fair coin was tossed an infinite ammount of times is it possible that heads or tails always comes up? Or is it necessary that both conditions will be realized. I did some initial calculations on this: Probability of always heads = (1/2^n) where n is the number of times flipped. lim(n->infinity) (1/2^n) = 0 Which would conclude that there is a 0 chance for heads to always occur over an infinite span? That both sides would necessary land? I have searched for an hour and couldn't find an answer to this. -Ryan
It sounds like you are giving me two different answers. No but yes. Is it possible for a fair coin to always land heads over an infinite span? -Ryan
Same result. "All heads" is one of an infinite number of possible outcomes - each of which has the probability 1/infinity of ocurring. One of them has to occur, but the probability that it will be the same as your prediction is zero (same story as with the probability of picking a given real number in a certain interval - there was a thread about this a while ago). Do the experiment to see this for yourself... BTW, Welcome (back?) to sciforums...
It depends by what you mean by "possible outcome." You've described an unusual scenario: If you toss an infinite number of coins there are an infinite number of possible outcomes. This gives each possibility a probability of 0. Yet if you toss an infinite number of coins, you have to get one of these outcomes - you'll just never be able to predict which one in advance. So if you toss an infinite number of coins, whatever the outcome is it won't be "all heads," "all tails," "head-tail-head-tail...," or any other sequence you care to list.
Warning! Warning Will Robinson! Talking about infinity and probability in the same sentence is a dangerous thing to do. You will almost always make an incorrect statement. Please Register or Log in to view the hidden image! You can talk about probabilities in an infinite domain, but not if you give each outcome an equal chance. You have to give weights to the outcomes so that the total weight is finite. The correct way to put it with your original question (I will ignore the faux pas of an infinite number of coin flips) is... "The probability of all heads approaches zero as the number of flips goes to infinity." Please Register or Log in to view the hidden image! Terandle's analysis was almost correct. I know this all may sound picky to non-math types, but believe me...in math the devil is in the details.
This I understand but... I am going to try and say this in a couple different ways as it is hard for me to put to words. Let there be a sequence of infinte length. The elements in this sequence are random with the same fixed number of possible outcomes (A fair coin was the random element in my example). Is it necessarily so that all possible outcomes of this random element will realize themselves over the infinite sequence? If so, is it necessarily so that it will happen an infinite ammount of times? -Another way- It makes sense to me that an infinite coin flips result in an infinite number of heads and tails. But is that necessarily so? Can the number of tails be finite or zero?
Yes. I think this is called a Poincare cycle. If the coin is fair, I think you'd get an infinite number of heads and tails. A result of a finite number of tails would be indistinguishable from no tails, in comparison to an infinite number of heads, so for the same reason I don't think it could happen.
Yes, James, you are correct. A finite number of tails would occur in a finite number of coin flips, say n. If you ignore the first n flips, you are left with the tail of the sequence, which is all heads. *If* it is correct that an infinite sequence of all heads is impossible, it then follows that there must be, in any random sequence, and infinity of both heads and tails. Please Register or Log in to view the hidden image!
This mathematical equation underlies the concept of what a limit is. Although the equation is true, it is interpreted as the limit of the expression .5^n, which is zero. However, .5^n never actually equals zero. It's a lot like a common mathematical physics lecture saying that 78% of the molecules of nitrogen gathers on this side of the room, and 21% of oxygen on the other, and perhaps all the argon in my desk. While there is practically no chance of it happening, mathematically it is possible. Does increasing the size of the room make it zero? Still, technically, no. This is analogous to applying limits in the coin toss. The technical answer would be no. But this is only what I think. You can try asking Marilyn vos Savant.
As long as n is finite. If n is inifinity, ie if you take the limit, then .5^n does actually equal zero. Unless the room is infinite in size. Then, technically, yes.
To be pernickity - isn't it 'limits to zero' or 'the limit equals zero', not 'equals zero'? There is no real number n such that .5^n = 0 Please Register or Log in to view the hidden image!
True but the extended real number line does define: r/infinity to equal zero, where r is a real number. But this isn't the point of my thread. Is all heads possible? So far we have JR- No, Facial- Yes
Given an infinite sequence, the probability of getting any particular number of heads is infinitesimal. At that point it's better to start working with proportions of heads and talking about probability density.
The usual definition of Probability asserts that P(Heads) = Limit(Heads/Flips) as Flips -> Infinity. So if a coin turns up heads an infinite number of times then it is, by definition, unfair. A fair coin can turn up heads any given finite number of times, but not an infinite number of times. I believe that it is similar with randomness, but I don't have a reference. A fair and random coin could turn up "heads, tails, heads, tails, ..." any given finite number of times and still be random, but if it did that an infinite number of times then it would, by definition, not be random. So my answer is, "No, by definition". -Dale
I'm beginning to think there are different kinds of zero probability. You have impossible events that'll never occur even if you repeat the experiment an infinite number of times, and you have zero probability events, where you expect the outcome zero times in any finite number of repetitions and at least once in an infinite number of repetitions. Since you can't repeat an experiment an infinite number of times, there's no practical difference and the only distiction is a conceptual one.
