# How many times do you have to flip a coin to ensure a heads **URL:** \ **Category:** Great Debates\ **Created:** [May 23, 2007, 11:06pm UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290 "2007-05-23T23:06:03Z")\ **Posts on this page:** 20\ **Page:** 5
**Author:** ![Hypnagogic\_Jerk](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hypnagogic_jerk/32/4252_2.png) [@Hypnagogic\_Jerk](https://boards.straightdope.com/u/Hypnagogic_Jerk)\ **Post date:** [May 24, 2007, 8:10pm UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/81 "2007-05-24T20:10:24Z")
[QUOTE=Polerius] (and, from what I know, mathematicians don’t consider Dirac Delta functions as “real functions”) [/QUOTE] Of course a Dirac delta isn’t a function, not in the classical sense anyway. If it were, it’d be a function whose support has Lebesgue measure zero and whose integral is nonzero, and as we know this is impossible. (Of course, these objects are still useful and we can use distribution theory to work with them formally.) Ah, physicists… 😛 Anyway, I agree with **Indistinguishable** : you can’t use probability theory to prove or disprove Intelligent Design, or the existence of God, since mathematics exists only in an idealized world anyway. It can be used to _model_ the real world with great success, but if you try to abuse it your model just breaks down. Better to concentrate on discussing probability itself. ---
**Author:** ![kanicbird](https://avatars.discourse-cdn.com/v4/letter/k/5f8ce5/32.png) [@kanicbird](https://boards.straightdope.com/u/kanicbird)\ **Post date:** [May 24, 2007, 10:01pm UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/82 "2007-05-24T22:01:14Z")
[QUOTE=Little Nemo] 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 [/QUOTE] Well no, it was not a attempt to prove God. It was to combat the atheistic value of the monkeys and the typewriter. If anything is it a attempt to prove that there is a real chance to absolutely prove atheism is incorrect. Note this is not the same as saying tring to prove atheism is incorrect, just that the chance exists via: 1 - Showing some step in life is impossible to happen naturally. This possibility still exists. 2 - Show that the probability of some step in life is too unlikely to actually happen. This has mainly been shot down with the circular dice mentioned above. But then again those dice are theoretical and our universe is on a discrete quantum level, so there are never a infinite # of points. Also someone asked me why I said that God will not allow us to scientifically prove Him ever. The simple answer is He told us this (in scripture), and He designed the universe, He’s not going to make a mistake. ---
**Author:** ![Shamozzle](https://avatars.discourse-cdn.com/v4/letter/s/9f8e36/32.png) [@Shamozzle](https://boards.straightdope.com/u/Shamozzle)\ **Post date:** [May 24, 2007, 10:04pm UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/83 "2007-05-24T22:04:21Z")
[QUOTE=Giles] 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. [/QUOTE] Yes, of course. I simply assumed we were talking in the theoretical. ---
**Author:** ![BrainGlutton](https://avatars.discourse-cdn.com/v4/letter/b/82dd89/32.png) [@BrainGlutton](https://boards.straightdope.com/u/BrainGlutton)\ **Post date:** [May 24, 2007, 10:29pm UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/84 "2007-05-24T22:29:44Z")
[QUOTE=Shamozzle] Any surface has an infinite number of points. [/QUOTE] Geometrically, yes. But the surface of any physical object is composed of a finite number of molecules. And IRL, there can be no such thing as perfectly spherical dice. ---
**Author:** ![BrainGlutton](https://avatars.discourse-cdn.com/v4/letter/b/82dd89/32.png) [@BrainGlutton](https://boards.straightdope.com/u/BrainGlutton)\ **Post date:** [May 24, 2007, 10:31pm UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/85 "2007-05-24T22:31:10Z")
[QUOTE=kanicbird] It’s a thought experiment. As I pointed out above I don’t expect it to prove the existance of God, as God will not allow that. [/QUOTE] Why would God not want his existence proven? ---
**Author:** ![begbert2](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/begbert2/32/3357_2.png) [@begbert2](https://boards.straightdope.com/u/begbert2)\ **Post date:** [May 24, 2007, 10:38pm UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/86 "2007-05-24T22:38:11Z")
[QUOTE=kanicbird] Well no, it was not a attempt to prove God. It was to combat the atheistic value of the monkeys and the typewriter. If anything is it a attempt to prove that there is a real chance to absolutely prove atheism is incorrect. Note this is not the same as saying tring to prove atheism is incorrect, just that the chance exists via: 1 - Showing some step in life is impossible to happen naturally. This possibility still exists. 2 - Show that the probability of some step in life is too unlikely to actually happen. This has mainly been shot down with the circular dice mentioned above. But then again those dice are theoretical and our universe is on a discrete quantum level, so there are never a infinite # of points. Also someone asked me why I said that God will not allow us to scientifically prove Him ever. The simple answer is He told us this (in scripture), and He designed the universe, He’s not going to make a mistake. [/QUOTE] 1 seems either highly unlikely or impossible, since our perception of what is naturally possible is built based on what we observe to have happened. If we found something that our current models don’t explain, we’d question our models, not scream “God did it!”. Oh, and you do realize that if you disprove atheism, then you’ve proven theism, right? If God’s in the market to avoid being proven to exist, he will not have done so sloppy a job in creating the universe as to allow the discovery of an impossible occurrence. So you of all people should believe that atheism can never be disproven, by this or any other method. 🙂 And you do realize that the “quanta on the sphere” in the analogy translates into “a finite number of possibilities” when translated to the general real-world applications, right? All that that would do would remove the case of “zero probability” possible events, by removing the infinity from the denominator, which makes everything easy: nonzero probabilites aren’t impossible, no matter how small, by definition. 2 is shot down. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [May 24, 2007, 11:05pm UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/87 "2007-05-24T23:05:20Z")
Incidentally, all the “spheres in the real world are quantized” stuff doesn’t really do anything to attack the fact that events of probability 0 are possible in the real world. Spherical dice were apparently an easily graspable way to communicate the situation, but they’re not the only example of the phenomenon. The coin flipping example itself demonstrates everything we need. Sit down and flip coins once a day from here on out; you’ll end up producing some infinite sequence of heads and tails (maybe HHHHHHH…, maybe HTHTHTHTHT…, maybe HHTHTHHTHTTTHHTHTHTHHTHT…, I have no idea, but you’ll produce _some_ sequence). But whatever infinite sequence you produce, it will be one which had probability 0 of being produced (whatever the sequence under consideration, you will have had probability 1/2^n of producing its first n ‘digits’; the probability of producing the whole thing must, therefore, be \<= 1/2, \<= 1/4, \<= 1/8, etc. The only nonnegative real number satisfying all of that is 0.). Thus, unavoidably, some situation of probability 0 ends up being the one that actually goes down, demonstrating that things of probability 0 aren’t always impossible. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [May 24, 2007, 11:17pm UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/88 "2007-05-24T23:17:00Z")
[QUOTE=Indistinguishable] Thus, unavoidably, some situation of probability 0 ends up being the one that actually goes down, demonstrating that **things of probability 0 aren’t always impossible.** [/QUOTE] Just to reword this bolded part, what I mean is that **not everything of probability 0 is impossible**. But, sure, some things _are_ impossible, and those things also have probability 0. Read all the above discussion for further information, offer void where prohibited, etc., … ---
**Author:** ![kanicbird](https://avatars.discourse-cdn.com/v4/letter/k/5f8ce5/32.png) [@kanicbird](https://boards.straightdope.com/u/kanicbird)\ **Post date:** [May 24, 2007, 11:27pm UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/89 "2007-05-24T23:27:51Z")
[QUOTE=BrainGlutton] Why would God not want his existence proven? [/QUOTE] In general terms I can answer that sort of: 1 - It is not part of His divine plan 2 - It is for His Glory The actual reason of how this fits into His plan to His glory is reserved for Him. [QUOTE=begbert2] 1 seems either highly unlikely or impossible, since our perception of what is naturally possible is built based on what we observe to have happened. If we found something that our current models don’t explain, we’d question our models [/QUOTE] Yes highly unlikely, but this thread has shown that even things highly unlikely things could happen. > [@](#): > > Oh, and you do realize that if you disprove atheism, then you’ve proven theism, right? If God’s in the market to avoid being proven to exist, he will not have done so sloppy a job in creating the universe as to allow the discovery of an impossible occurrence. So you of all people should believe that atheism can never be disproven, by this or any other method. Yes, this seems to be the case. > [@](#): > > And you do realize that the “quanta on the sphere” in the analogy translates into “a finite number of possibilities” when translated to the general real-world applications, right? All that that would do would remove the case of “zero probability” possible events, by removing the infinity from the denominator, which makes everything easy: nonzero probabilites aren’t impossible, no matter how small, by definition. Well not so fast here. While this may be true for a infinite time, and it helps to have infintie space, we have neither infinite time or space, so not everything that could happen will, but like those dice, something will happen. ---
**Author:** ![kanicbird](https://avatars.discourse-cdn.com/v4/letter/k/5f8ce5/32.png) [@kanicbird](https://boards.straightdope.com/u/kanicbird)\ **Post date:** [May 24, 2007, 11:34pm UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/90 "2007-05-24T23:34:13Z")
[QUOTE=Indistinguishable] Incidentally, all the “spheres in the real world are quantized” stuff doesn’t really do anything to attack the fact that events of probability 0 are possible in the real world. [/QUOTE] Actually it does. If all that is available is whole numbers on this sphere due to quantum effects you can never have a value of 2.5 , it is a impossibility as opposed to a probability approaching zero. ---
**Author:** ![begbert2](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/begbert2/32/3357_2.png) [@begbert2](https://boards.straightdope.com/u/begbert2)\ **Post date:** [May 24, 2007, 11:47pm UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/91 "2007-05-24T23:47:42Z")
[QUOTE=kanicbird] Yes highly unlikely, but this thread has shown that even things highly unlikely things could happen. [/QUOTE] Yes, but pardon me if I don’t hold my breath. For this to happen science would have to “give up” and stop searching for rational explanations, which isn’t the way science generally works. [QUOTE=kanicbird] Well not so fast here. While this may be true for a infinite time, and it helps to have infintie space, we have neither infinite time or space, so not everything that could happen will, but like those dice, something will happen. [/QUOTE] No, really, a non-zero probability means that the event isn’t impossible. It doesn’t mean that it _will_ happen, but it remains possible as long as the probability remains nonzero, and (as has been shown) sometimes even if it doesn’t. This works in finite spaces and times as well as infinite ones. And when you’re talking about something that already occurred (like life, the universe, and everything you might base an anti-atheist argument on) the question of whether something might possibly happen is moot: it happened; ergo it was possible. You’re goal would be to prove that it would have been impossible to do _without god_; this is a question of causes, not probability. Once you understand that incredibly improbable events can indeed happen (since _something_ has to happen), probability has no more place in the discussion. ---
**Author:** ![BrainGlutton](https://avatars.discourse-cdn.com/v4/letter/b/82dd89/32.png) [@BrainGlutton](https://boards.straightdope.com/u/BrainGlutton)\ **Post date:** [May 25, 2007, 12:11am UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/92 "2007-05-25T00:11:23Z")
[QUOTE=kanicbird] In general terms I can answer that sort of: 1 - It is not part of His divine plan 2 - It is for His Glory The actual reason of how this fits into His plan to His glory is reserved for Him. [/QUOTE] Are you basing that on Scripture, or what? ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [May 25, 2007, 12:36am UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/93 "2007-05-25T00:36:30Z")
[QUOTE=kanicbird] Actually it does. If all that is available is whole numbers on this sphere due to quantum effects you can never have a value of 2.5 , it is a impossibility as opposed to a probability approaching zero. [/QUOTE] Read the rest of the post you are replying to. We may lose the sphere example, but the sphere example wasn’t essential; the coin example is good enough in itself. ---
**Author:** ![kanicbird](https://avatars.discourse-cdn.com/v4/letter/k/5f8ce5/32.png) [@kanicbird](https://boards.straightdope.com/u/kanicbird)\ **Post date:** [May 25, 2007, 12:40am UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/94 "2007-05-25T00:40:50Z")
[QUOTE=BrainGlutton] Are you basing that on Scripture, or what? [/QUOTE] Yes [QUOTE=begbert2] the question of whether something might possibly happen is moot: it happened; ergo it was possible. [/QUOTE] Well we know that life is possible, the question is if some intelligence was needed to design it or if the whole thing could come about naturally. ---
**Author:** ![begbert2](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/begbert2/32/3357_2.png) [@begbert2](https://boards.straightdope.com/u/begbert2)\ **Post date:** [May 25, 2007, 12:47am UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/95 "2007-05-25T00:47:39Z")
[QUOTE=kanicbird] Well we know that life is possible, the question is if some intelligence was needed to design it or if the whole thing could come about naturally. [/QUOTE] Yep. Of course, I believe I already know the answer to that question - if all you’re asking is if it’s _possible_ that it happened naturally. ---
**Author:** ![Little\_Nemo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/little_nemo/32/3120_2.png) [@Little\_Nemo](https://boards.straightdope.com/u/Little_Nemo)\ **Post date:** [May 25, 2007, 4:13pm UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/96 "2007-05-25T16:13:47Z")
Kanicbird, what you’re trying to do is the equivalent of arguing that dinosaurs or dragons exist and then claiming that by demonstrating dinosaurs probably don’t exist, you’ve proven that dragons are real. Sorry but it’s a flawed argument - you need to prove the existence of dragons on its own merits not by tying it in to the issue of dinosaurs. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [May 25, 2007, 4:35pm UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/97 "2007-05-25T16:35:54Z")
[QUOTE=Little Nemo] Kanicbird, what you’re trying to do is the equivalent of arguing that dinosaurs or dragons exist and then claiming that by demonstrating dinosaurs probably don’t exist, you’ve proven that dragons are real. Sorry but it’s a flawed argument - you need to prove the existence of dragons on its own merits not by tying it in to the issue of dinosaurs. [/QUOTE] Huh? That would be a perfectly valid argument for “Dragons probably exist”, if one could actually pull off both steps. I think **kanicbird** ’s efforts here are misguided, but not because they happen to tie one issue to another; that’s hardly a logical sin. ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [May 25, 2007, 7:23pm UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/98 "2007-05-25T19:23:56Z")
[QUOTE=kanicbird] Statement B has been mathematically proven, called the law of large numbers. > **[Law of large numbers](https://en.wikipedia.org/wiki/Law_of_large_numbers)** > > In probability theory, the law of large numbers is a mathematical law which states that the average of the results obtained from a large number of independent random samples converges to the true value, if it exists. More formally, the law of large numbers states that given a sample of independent and identically distributed values, the sample mean converges to the true mean. The law of large numbers is important because it guarantees stable long-term results for the averages of some random event... [/QUOTE] If I’m reading that link correctly, what is proved as the "Strong Law of Large Numbers) only says that the quarter will “almost surely” land on heads and tails an equal number of times. But “almost surely” is a technical term, with its own [wikipedia article](http://en.wikipedia.org/wiki/Almost_surely), and at that article, we find that “almost surely” is compatible with the possiblity that it might not happen, _even if_ its probability is 1. -FrL- ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [May 25, 2007, 7:28pm UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/99 "2007-05-25T19:28:06Z")
Oops. Boy was I late to this party. Sorry for my previous post! 🙂 -FrL- ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [May 25, 2007, 7:33pm UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/100 "2007-05-25T19:33:03Z")
[QUOTE=Indistinguishable] So, just to head it off, yes, two events can have the exact same probability, even though one is possible and the other is impossible. Probabilities aren’t fine-grained enough to make a distinction. Same as two regions can have the same area, even if one has more points than the other (e.g., the area of the border of India is 0, same as the area of the North Pole and of the empty set, even though there are more points on the border of India). If that bothers you, feel free to develop an alternative theory of probability (or of area, for that matter) to more accurately capture your own conception of it; I even support such efforts, but, I’ll warn you, there are severe difficulties in trying to get around this particular aspect of the conventional theory. [/QUOTE] Wait. More points? I thought all line segments, lines, curves and areas had the same number of points. -FrL- [Previous page](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290.md?page=4) [Next page](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290.md?page=6)