# How many times do you have to flip a coin to ensure a heads
**URL:** https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290
**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:** 1
### Author:  [@kanicbird](https://boards.straightdope.com/u/kanicbird)
#### Post date: [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/405290/1 "2007-05-23T23:06:03Z")
Believe it or not this comes from all the talk about theistic and atheistic faith issues that have come up lately.
Here is the premises:
If you flip a infinite number of coins a infinite number of times you will get a infinite number of coins that will never come up heads.
-but-
If you flip any one coin a infinite number of times it will hit heads and tails equally.
These 2 can’t coexist.
There is a saying that if you take a room full of monkeys and give them a typewriter and a infinite amount of time you will eventually get a copy of Hamlet. No we all know this could never happen either as monkeys will destroy the paper at a much higher probability then completing a single letter (barring evolution).
There seems like there is some limit on probability, when the chance of something happening is so so so small it is not just effectively zero it _ **is** _ zero and some things just can never happen\*.
- Now this is where the concept of God comes in, things like creation and life may have zero chance of happening on there own without a ‘God’.
(Not that I’m hanging my theists hat on this, as I don’t think God designed the system for Him to be proven, but I would like to see the responses.)
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### Author:  [@Mikemike2](https://boards.straightdope.com/u/Mikemike2)
#### Post date: [May 23, 2007, 11:11pm UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/2 "2007-05-23T23:11:56Z")
If statement A is correct then statement B is incorrect. The coin flipped in statement B could be one of the infinite amount of coins that will never come up heads. So there is no problem with both of these statements coexisitng. We just have a hard time wrapping our heads around how big a number infinite is 🙂
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### Author:  [@begbert2](https://boards.straightdope.com/u/begbert2)
#### Post date: [May 23, 2007, 11:16pm UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/3 "2007-05-23T23:16:14Z")
I completely don’t get your point.
A coin can be fair (that is, have an equal intrinsic probability of turning up heads or tails on each successive flip) and still land tails-up on each individual flip for any finite\* number of coin flips. There’s no contradiction here; the odds of it happening get infinitesimally small as the sequence of flips gets long, but the occurrence of a heads never becomes impossible (assuming a fair coin). It also never becomes certain. If it’s a fair coin, it’s even odds each flip, no matter what’s happened before.
Now, as I flip a coin and each time it comes up tails, I will gradually begin to doubt that it’s a fair coin, since it doesn’t seem to be behaving the way I’d _expect_ a fair coin to act. But no amount of flipping will _prove_ that a heads is impossible, and no number of flips is enough to _guarantee_ that a heads will show. That’s not the nature of coin flips.
So out of curiousity, how does this relate to theology?
- It is of course impossible to to flip a coin an infinite number of times. Don’t toy with him, **Mikemike2**.
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### Author:  [@Nancarrow](https://boards.straightdope.com/u/Nancarrow)
#### Post date: [May 23, 2007, 11:17pm UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/4 "2007-05-23T23:17:42Z")
[QUOTE=kanicbird]
Believe it or not this comes from all the talk about theistic and atheistic faith issues that have come up lately.
Here is the premises:
If you flip a infinite number of coins a infinite number of times you will get a infinite number of coins that will never come up heads.
[/QUOTE]
I’m pretty sure that’s incorrect. Because you’re flipping them an infinite number of times, NONE of them would never come up heads.
If you just flipped them a finite number of times, no matter how many that is, then sure, an infinite number of them would never come up heads.
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### Author:  [@kanicbird](https://boards.straightdope.com/u/kanicbird)
#### Post date: [May 23, 2007, 11:19pm UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/5 "2007-05-23T23:19:45Z")
[QUOTE=Mikemike2]
If statement A is correct then statement B is incorrect. The coin flipped in statement B could be one of the infinite amount of coins that will never come up heads. So there is no problem with both of these statements coexisitng. We just have a hard time wrapping our heads around how big a number infinite is
[/QUOTE]
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...
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### Author:  [@BrainGlutton](https://boards.straightdope.com/u/BrainGlutton)
#### Post date: [May 23, 2007, 11:26pm UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/6 "2007-05-23T23:26:21Z")
[QUOTE=kanicbird]
- Now this is where the concept of God comes in, things like creation and life may have zero chance of happening on there own without a ‘God’.
[/QUOTE]
What you’re addressing here is the basic unanswered question of cosmology: “Why is there something rather than nothing?” But invoking God does **not** answer that question. If God exists, God is something. Why does God exist rather than not existing? (And please **don’t** waste our time with the [ontological argument!](http://en.wikipedia.org/wiki/Ontological_argument) It’s bullshit!)
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### Author:  [@kanicbird](https://boards.straightdope.com/u/kanicbird)
#### Post date: [May 23, 2007, 11:26pm UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/7 "2007-05-23T23:26:35Z")
[QUOTE=begbert2]
So out of curiousity, how does this relate to theology?
[/QUOTE]
The contention that certain things like creation & life are too small probabilistically to happen on their own (with out a ‘designer’), they fall below a quantized threshold that makes things with such a small chance of happening actually zero chance of happening.
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### Author:  [@begbert2](https://boards.straightdope.com/u/begbert2)
#### Post date: [May 23, 2007, 11:32pm UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/8 "2007-05-23T23:32:04Z")
Oh, I get it. The answer is that “really improbable” does not mean “impossible”, no matter _how_ improbable it is.
Plus, well, really improbable things become more probable with the amount of _tries_ you have at them (assuming all you’re fighting against is probability). Given how many suns and maybe planets there are, and how long the universe has been around, hasn’t there been enough chances even for some really improbable things to occur, even if only once or twice?
(Plus maybe we’re not the only universe ever to happen. We dunno.)
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### Author:  [@Der\_Trihs](https://boards.straightdope.com/u/Der_Trihs)
#### Post date: [May 23, 2007, 11:36pm UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/9 "2007-05-23T23:36:30Z")
[QUOTE=kanicbird]
The contention that certain things like creation & life are too small probabilistically to happen on their own (with out a ‘designer’), they fall below a quantized threshold that makes things with such a small chance of happening actually zero chance of happening.
[/QUOTE]
Except there’s no evidence of that. Scientists who try cooking up life’s precursors in simulations of Earth’s early environment find it quite easy. From what I’ve read such arguments usually try to talk about a large, complicated DNA molecule or even a cell coming together by pure chance; not some simple molecules becoming self replicating, and only after much evolution becoming life. Also, they often ignore evolution’s abilty to weed out alternates, and ignore the natural tendancy of atoms and molecules to order themselves in various ways - something life exploits a LOT.
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### Author:  [@Mikemike2](https://boards.straightdope.com/u/Mikemike2)
#### Post date: [May 23, 2007, 11:37pm UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/10 "2007-05-23T23:37:06Z")
[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]
I did say “if”. 🙂 I believe in the law of large numbers, but as long as were are dealing with the term infinite, there can be no answers as there is no point at which to measure.
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### Author:  [@Larry\_Borgia](https://boards.straightdope.com/u/Larry_Borgia)
#### Post date: [May 23, 2007, 11:39pm UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/11 "2007-05-23T23:39:26Z")
You’ll never have an infinite number of coins, nor will you be able to flip a coin an infinite number of times, so these paradoxes are irrelevant for the real world.
To expect a run of n heads you’ll need 2^n coins. Flipping 1024 coins ten times each will get you one run of heads. Or rather, if you have a large number of trials flipping 1024 coins ten times each you’ll average one all-head run per trial. No matter how large your all-head run needs to be, there is a much larger number of coins which will make your all-head run probable.
In order to determine the likelihood of an improbable event we need a clear understanding of just how improbable it is. We don’t know that with life, but given the age and size of the universe, I’d bet the probability of life is pretty high. (We do know it isn’t zero, at least.)
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### Author:  [@begbert2](https://boards.straightdope.com/u/begbert2)
#### Post date: [May 23, 2007, 11:44pm UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/12 "2007-05-23T23:44:55Z")
[QUOTE=Larry Borgia]
(We do know it isn’t zero, at least.)
[/QUOTE]
Actually the theistic angle on the OP _does_ assume it’s naturally impossible -thus making God (or something like it) necessary. If it’s merely unlikely, God is unnecessary.
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### Author:  [@The\_Hamster\_King](https://boards.straightdope.com/u/The_Hamster_King)
#### Post date: [May 23, 2007, 11:49pm UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/13 "2007-05-23T23:49:32Z")
[QUOTE=kanicbird]
If you flip a infinite number of coins a infinite number of times you will get a infinite number of coins that will never come up heads.
[/QUOTE]
The phrase “never come up heads” is meaningless if you flip a coin an infinite number of times.
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### Author:  [@kanicbird](https://boards.straightdope.com/u/kanicbird)
#### Post date: [May 23, 2007, 11:55pm UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/14 "2007-05-23T23:55:19Z")
[QUOTE=Larry Borgia]
You’ll never have an infinite number of coins, nor will you be able to flip a coin an infinite number of times, so these paradoxes are irrelevant for the real world.
[/QUOTE]
Yes, and the universe is finite and had a limited amount of time available to develop life - so there is less chance of a complex thing happening if there is finite space and time then infinite space and time.
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### Author:  [@kanicbird](https://boards.straightdope.com/u/kanicbird)
#### Post date: [May 23, 2007, 11:58pm UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/15 "2007-05-23T23:58:15Z")
[QUOTE=Larry Borgia]
You’ll never have an infinite number of coins, nor will you be able to flip a coin an infinite number of times, so these paradoxes are irrelevant for the real world.
[/QUOTE]
Yes, and the universe is finite and had a limited amount of time available to develop life - so there is less chance of a complex thing happening if there is finite space and time then infinite space and time.
If it can be proven that life (not to mention ‘creation’ itself) is impossible given infinite time and space, I would contend that it is also impossible with finite time and space
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### Author:  [@Kimstu](https://boards.straightdope.com/u/Kimstu)
#### Post date: [May 23, 2007, 11:59pm UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/16 "2007-05-23T23:59:31Z")
Remember also that the physical universe as we understand it was never “trying” to produce any particular form of life, much less the one that happened to evolve.
It’s like picking up a bridge hand of 13 cards dealt randomly from a pack of 52. For **any particular hand** , the odds of being dealt **exactly that hand** at random are about 1 in 400000000000000000000. (Yes, 20 zeroes. :eek: )
**That’s orders of magnitude bigger than the number of seconds in the lifetime of the universe.** Which means that if you were dealing one bridge hand per second from the birth of the universe to the present moment, the odds would still be **thousands to one against** your ever getting **that particular hand**.
And yet, that’s the hand that was dealt. Does this demonstrate statistically that there must be an Intelligent Designer of bridge hands who voluntarily stepped in and gave you that particular hand, rather than random chance?
By no means. The thing is that if you’re dealing bridge hands at random, you’re going to deal out **some** configuration of 13 cards each time. Each random configuration is just as unlikely as any of the others.
Similarly, the current precise configuration of the physical universe, including this particular set of carbon-based life forms existing on this particular planet orbiting this particular star, is indeed massively improbable. But then, **so are all the other possible random configurations of the physical universe**.
It would indeed be unrealistically unlikely that the universe could have evolved into any one _pre-selected_ configuration purely at random. But there was no pre-selection involved. The universe could have happened to produce some other other kind of self-aware life form on a different planet near a different star; and if it had (or has), those other life forms would/will probably be just as awed as we are at the mind-boggling odds-beating improbability of their very existence. That doesn’t mean that there is any statistical reason to think that anything but chance and physical laws were involved in the process.
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### Author:  [@The\_Hamster\_King](https://boards.straightdope.com/u/The_Hamster_King)
#### Post date: [May 23, 2007, 11:59pm UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/17 "2007-05-23T23:59:50Z")
[QUOTE=kanicbird]
There seems like there is some limit on probability, when the chance of something happening is so so so small it is not just effectively zero it _ **is** _ zero and some things just can never happen\*.
[/QUOTE]
I raised this issue in February in [this thread](http://boards.straightdope.com/sdmb/showthread.php?t=407751&highlight=bowling) about improbable configurations of matter in an infinite universe.
> [@](#):
>
> - Now this is where the concept of God comes in, things like creation and life may have zero chance of happening on there own without a ‘God’.
In order to make such an argument you need to determine the probability of either event occurring spontaineously. Just saying “well gosh it SEEMS really, really unlikely” doesn’t cut it.
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### Author:  [@begbert2](https://boards.straightdope.com/u/begbert2)
#### Post date: [May 24, 2007, 12:02am UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/18 "2007-05-24T00:02:56Z")
[QUOTE=kanicbird]
If it can be proven that life (not to mention ‘creation’ itself) is impossible given infinite time and space, I would contend that it is also impossible with finite time and space
[/QUOTE]
Sure. But it hasn’t been so proven, now has it? Doesn’t that make this something of a strawman?
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### Author:  [@kanicbird](https://boards.straightdope.com/u/kanicbird)
#### Post date: [May 24, 2007, 12:10am UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/19 "2007-05-24T00:10:34Z")
[QUOTE=Pochacco]
The phrase “never come up heads” is meaningless if you flip a coin an infinite number of times.
[/QUOTE]
This is correct, I just realized my initial premise A was in error, the number of coins left converges on zero much faster then the number of flips approaching infinity, so yes you won’t have a single coin that fits that.
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### Author:  [@The\_Hamster\_King](https://boards.straightdope.com/u/The_Hamster_King)
#### Post date: [May 24, 2007, 12:14am UTC](https://boards.straightdope.com/t/how-many-times-do-you-have-to-flip-a-coin-to-ensure-a-heads/405290/20 "2007-05-24T00:14:02Z")
Here’s a quote from \*\*severus \*\* in the previous thread that may be enlightening:
[QUOTE=severus]
When we’re dealing with an uncountable probability space, saying that an event happens with probability 1 (we also say it happens “almost surely”) doesn’t mean that it must happen. Similarly, saying that an event happens with probability 0 does not mean that it cannot happen.
For example, pick a number at random from the set of reals between 0 and 1. What is the probability of you picking exactly 1/2? The probability is 0. But it doesn’t mean that this cannot happen. Similarly, the probability of you picking something else than 1/2 is 1, but as we can see, this isn’t what will necessarily happen.
This is related to what we know in popular culture as the infinite monkey theorem. If you have a monkey typing at random on a typewriter, for an infinite amount of time, any given finite string of text will be found somewhere within the output of your typing monkey with probability 1. However, it doesn’t mean that it must be found in the monkey’s output. After all, the monkey could very well only be typing the letter ‘a’. The probability of this happening is 0, of course, but it is a valid result. This is because the space of all infinite sequences of letters is uncountable.
[/QUOTE]
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