# Is it possible to flip a coin an infinite number of times and never land on tails?

If I would flip a coin an infinite number of times would it be possible to never land on tails? In other words if there's an infinite number of chances of something happening is it still possible for it not to happen?

• The probability of getting 0 tails in N trials goes to zero as N increases. Probability doesn't deal well with actual infinities, thoiugh, because the sample space would be the set of all (countably) infinite coin tosses, which is itself uncountably infinite. The math gets a hernia trying to lift that load, and dies a miserable death. Jul 13, 2020 at 17:51
• Why couldn't it happen? One flip has no knowledge of any other flip. It's true that if you flip infinitely many coins that the probability they're all heads is zero. But in infinitary probability theory, events with probability zero may still happen. Jul 13, 2020 at 19:37
• This is a math question, not a philosophical one. Also the question should be precise about whether this is a fair coin, and whether the coin is ordinary, as opposed to a coin having heads on both sides. Jul 14, 2020 at 3:32
• Jul 14, 2020 at 4:03

Yes.

In fact, having your (fair) coin turn up heads every single time is as likely as every other possible permutation.

Then, since there is literally an infinite number of sequences, that's not very likely.

• What other definition of "impossible" could there be, other than that the probability of occurrence is literally zero? Not tiny, not infinitesimal, but literally equal to zero? Jul 21, 2020 at 18:33

No, it's not possible.

But it really depends by what you mean by 'infinite', 'never' and 'possible'.

Considering a person (or a finite number of people [1]) flipping a coin at regular interval, what we can say is that the probability of only heads converge to 0 awfully fast.

P(all heads) = 1/2^(toss)

For a single person taking their sweet time and tossing every 5 second, after one hour it's about 1/(10^216). We estimate about 10^80 atoms in the whole universe, so that's about the same chance than picking the right atom out of the universe, 3 times in a row (give or take the probability to simultaneous win the lottery, be struck by lightning, get married and eaten by a shark [2]). Let's call that unlikely.

But for any finite number of tosses, it's never exactly P=0.

For any 'infinite' number of tosses, we can agree to say that the probability to have all heads is the same thing than the value P(n tosses) converges toward as the integer n grows to infinity, aka 0.

Wether you want think 'possible' means an exact 0, or wether you're satisfied with a convergent series limit is up to you.

Notes:

1. That might work with a countably infinite number of people, not sure.
2. I didn't compute that. YMMV.
• The probability of any event happening at a given time is also zero, but I would not say that this means that for all practical purposes there are no events ever happening. You probably meant that the frequency (real world occurrence) of a large number of flips landing on heads only is so scarce that it is equally valid to say that it does not happen at all. Which leads to the question how useful it is to think about real world occurrence when it comes to infinity... Jul 16, 2020 at 5:51
• It's really a math questions, but there are number of issues with the way it's formulated that make it hard to answer. Never implies time. But you can't flip an infinite number of coins without infinite time, etc. That said, I'm don't think the argument that anything happening at a given time is compelling (or the problem here). I'll edit my answer.
– ptyx
Jul 16, 2020 at 20:19

The probability of that event is zero. But that's a bit different from "cannot happen". For example, the probability of a dart landing on any particular point on a dartboard is zero, but it obviously has to land on one of them. And of course it would take an infinite amount of time to occur if it did, which bears the same relationship with "never" that "zero probabilty" has with "impossible".

• " And of course it would take an infinite amount of time to occur if it did," -- Of course this is only a conceptual question so time and physics aren't involved. We could just as easily flip infinitely many coins all at once! Jul 20, 2020 at 0:36
• "the probability of a dart landing on any particular point on a dartboard is zero" This assumes the board is continuous rather than made of atoms with finite positions. And our own infinite capacity to distinguish between positions. And violates unitarity, the modelling principle that the total probabilities add up to 1, there will be an outcome. Jul 20, 2020 at 7:00

If I would flip a coin an infinite number of times would it be possible to never land on tails?

Anything is possible when there are no constraints, so this is not a useful question, as it translates to: "Is it possible for a possible event to happen?".

In other words if there's an infinite number of chances of something happening is it still possible for it not to happen?

An infinite sequence of fair coin tosses contains any finite sequence of coin tosses with probability one (almost guaranteed). However, the number of tosses required to observe a series of length n grows so quickly that for practical purposes, even finite series of certain lengths become practically impossible to expect within reasonable time.

However, an infinite sequence of coin tosses can not contain every infinite series of coin tosses. Simple proof: If it did, then there would be an infinite series of coin tosses containing both an infinite sequence of heads and an infinite sequence of tails. But both cannot fit into one series (and if we could use both directions, we still cannot fit a third infinite series like HTHTHTHT...). So there is no guarantee of all infinite series occurring in a given infinite series of tosses. (And thus, of course, there is also no guarantee for an infinite series of heads to happen.)

• Doesn't answer the question. Jul 15, 2020 at 6:10
• @user4894: huh? First question is answered with yes/obviously. Second question answered with no, and proof. Jul 15, 2020 at 13:01

This sequence is just as likely as any other sequence. It CAN happen. It IS possible. However, the probability of it happening is 0.

As each sequence is equally likely (and total sequences are infinite), sum of probabilities of each sequence has to be one, this tells you that you need, heuristically,

[Probability of any sequence]*(total sequences)=1, or

[Probability of any sequence]*(infinity)=1

Therefore, the probability is 'constrained' to be 0. Any other number, and LHS becomes infinite.

Now you may wonder, how can 0*infinity be 1? This is something you encounter a lot in math, where you're summing up things that are infinitesmally small, but there are an infinite number of them that get summed up. What can the resulting sum be? Turns out, it can be anything from negative infinity to infinity, and that is where one starts to lose layman's intuition.

Think of it like this - if you add inifinitesmally small 'drops of water' into a big ocean, the ocean volume will not increase unless you add an infinite number of them. Then, the volume increase may be 0, or positive, or infinite! This depends on the interplay between the size of each drop and the number of drops added.

Although 'probability' is an intuitive concept, rigorously, it is just a mapping from the set of events to [0,1]. (i.e., probability is a 'machine' that inputs an event, and outputs a number in [0,1], and we interpret the output as 'probability of the input happening'). It is a function that by construction obeys sum laws (such as, total probability is always 1), it is from these laws that we find out the probability of things. So simply invoking these laws takes us to the answer that the probability of this event is 0. Still it can happen, but to be consistent with 'what probability means mathematically', we have to assign 0 probability to this event. So 0 probability is not synonymous with 'impossible'.

• I would concur that we get probabilities from two different sources: Laplacian probability and probability derived from frequency. Laplacian probability has to assume certain events as possible, for example a heads only sequence, even if there may be no real world example (or metaphysical possibility). I think one of the problems speaking of possibilities is being too quick with going from mathematical (logical) to metaphysical possibility. The former is an assumption. Jul 16, 2020 at 5:58

Whether any infinity at all is truly real, is very much a contentious and unsettled question - any answer depends on substantial ontological and epistemic framing, and is really only meaningful in relation to a particular set of that framing.

I suggest that in exactly the case of the logical concept of infinity, and only there, the abstract model of the coin cannot land only on one of it's sides - but any real coin flipped a real possible number of times can. Infinity is the definition of where the odds of an abstract model, approach it's true average behaviour.

Options like landing on an edge are not included in the model, but as the number of flips of a real coin a real number of times goes up it becomes not just possible but certain it will do so.

If we go to the universe for an answer, and look at a 'pure' coin, like a quantum state in the Schroedingers Cat thought experiment, the chance of a given outcome, finding the cat dead, can be modelled by a time related variable, and it could as far as we know outlast the universe. But only if it is kept completely isolated down to the last stray photon (or maybe gravity wave), unobserved, can the outcome remain undecided, and the quantum model continue to apply. In a quantum sense, it could only possibly be in an infinitely unlikely state by literally never being observed.

The 2nd law of thermodynamics and the arrow of time, seem to suggest it may not be simply that decreases of entropy are unlikely, but that consciousness is the taking of information from the environment and integrating that with what else is already known, and that of necessity pushes in the direction of increasing entropy, of pure states becoming mixed states, of the diffusion of information. Perhaps a reversal of entropy is possible, and a reversal in time, it could only be experienced by us in the conventional direction..? That would be where time is emergent from another way of picturing the world like probability space, and our awareness moves along contours on a landscape, with the principle of least action held firmly with one path along that axis, but the increase of entropy held 'fuzzily', with a set of probabilities up to 'no increase of entropy' and down to maximum entropy (interior of a blackhole) that our consciousness rather than the landscape itself requires. This is with a picture that takes conservation of information as real, to have entropy change not just stationary but reversing, the state information would have to become more isolated, the reverse of an observation.

The question has two parts: 1) Is it possible to flip a coin an infinite number of times, AND 2) never land on tails?

The answer to the first part is No. It's not possible to perform any action an infinite number of times.

Since the two parts are joined by AND, both parts would have to have the same answer in order that the entire question have any valid answer. Since the answer to the first half is no, the answer to the entire question is no.

Also, since the first part is not possible, the second part doesn't make sense in the context of the first part.

Added content: Some here don't like my crying foul on the question, with the excuse that, well no one really takes it as really meaning an "infinite" number of flips. My defense is simply, if the question can be so easily interpreted without the strict meaning of "infinite," then I suggest rephrasing it with what you really mean. For instance, "a sufficiently large number," or "Is there a large enough number,.." I contend that the use of the word "infinite" is problematic and creates logical conundrums and confusion in addressing what the questioner really wants to know. What does the questioner want to know? If it's so obvious that what he/she means is something other than "infinite," then say it.

• I don't think anyone means to imply that one could flip a coin infinitely many times in reality. The question is about the probability of randomly picking a certain bitstring out of the (abstract, conceptual) space of all possible bitstrings. Jul 20, 2020 at 22:03