In the beginning ...

[thestoat]

While this is a true statement it is not applicable to a situation where the outcome is purely random (ie monkeys or dice)

It was in the context of monkeys that this statement was made. I am wondering if the contradiction is a theoretical analogy to wave particle duality in light where some experiments show light is a wave and not a particle but other experiments show the opposite.

[Sean]

I think that the nature of light has got nothing to do with this and will only serve to cloud what is at heart a simple issue.

Just to go back to something I missed earlier:

I agree that the source does appear to contradict itself (so interesting that this is the first time you even acknowledge the source - before this you had metaphorical fingers in your ears saying "nah nah I can't hear you"). But I actually thin they are trying to be clever with maths, to sort of point holes in the infinity concept and also enthuse the reader.

So can Dr Math be dismissed as a reputable and reliable source? Is he is trying to be clever and enthuse the reader or stating mathematical fact?

I suspect we won't agree. There is conflicting evidence on the web and indeed within the mathematics. I would love someone with a deep knowledge of maths to explain one way or another but it does appear that even maths is split on this issue.

I have yet to find any conflicting evidence (with the exception of Dr Math who contradicts himself) from any reputable source on the web or within the mathematics. As for maths being split on the issue... The Infinite Monkey Theorem states that "a monkey typing random letters on a typewriter for an infinite amount of time will almost surely, as part of its output, produce the complete works of Shakespeare."

In the context of this debate the key words would have to be "almost surely" would they not? You can find the theorem on thousands of websites. I have not yet found a definitive wording (there are slight variations) but the all include the key words "almost surely".

I'll give the last word here to a couple of blokes who probably know a tad more about these things than any of us... or indeed all of us put together and fed to the monkeys!

Charles Kittel andHerbert Kroemer: "The probability of Hamlet is therefore zero in any operational sense of an event…", and the statement that the monkeys must eventually succeed "gives a misleading conclusion about very, very large numbers."
(Kittel, Charles and Herbert Kroemer (1980). Thermal Physics (2nd ed.). W. H. Freeman Company. pp. 53. ISBN 0-7167-1088-9.)



Mind you, I'm not sure how reliable Kittel is... The simpleton hasn't even won a Nobel prize!

[thestoat]

I actually have the book. But of course they talk about "very large numbers", not infinity. The reason I wonder if the nature of maths is split is because many out there regard zero probability as meaning impossible by its very nature. Yes Dr Math does appear to contradict itself - but I have not found anywhere on the site to suggest the same person answers all questions - it could well be there are 2 mathematitians there who disagree. I am asking them now why the discrepancy and will post back the result if/when I get it.

(And I have to confess I never understood Kittel's book - that subject was a nightmare to me)

[thestoat]

As I thought, there is more than one person answering the questions, so it could WELL be 2 mathematicians disagree.

We currently get about 200-300 questions per day, which are answered by volunteers, all of whom have other jobs

[Sean]

Cheers, much appreciated!

Actually, Kittel & Kroemer are talking explicitly about the Infinite Monkey Theorem in that quote... which is, I believe, what we are discussing.

[MajGenl.Meade]

*Which as any fule no means Quod Erat Demonstrandum**

Good to see Molesworth entering the lists!

[Sean]

lol - I knew somebody would appreciate that General...

[thestoat]

The thread that just won't die.

I had a response from Dr Maths and I append it here in its entirety, along with my original question to them for reference. I think the important comment is "Both statements are correct and they do not contradict" - so the unambiguous "if you have only a finite number of outcomes and you take an infinite number of trials, you will end up getting the outcome you are looking for" does help my argument.

I have asked for further clarification, but I *think* that the statement we (all?) thought contradictory maybe highlights our limit of maths understanding, since now they talk about limits rather than "reaching infinity". Anyway, I shall post back if and when I get an answer.



> [Question]
> >I am having a long argument about the odds of throwing a 6 using a
> >standard dice in infinite time. Your site appears to contradict itself
> >and this has added to the confusion.
> >
> >Here (http://mathforum.org/library/drmath/view/55871.html) you state
> >"if you have only a finite number of outcomes and you take an infinite
> >number of trials, you will end up getting the outcome you are looking
> >for".
> >
> >Here (http://mathforum.org/dr.math/faq/faq.prob.intro.html) you state
> >"Consider the example of picking a random number between 1 and 10 -
> >what is the probability that you'll pick 5.0724? It's zero, but it
> >could happen". Please help
> >
> >[Difficulty]
> >It appears the 2 statements on your web site contradict each other,
> >and this I find confusing.
> >
> >My feeling is that if a dice was rolled an infinite number of times
> >and a 6 did not show, then it was never possible to roll a six.
> >However, others believe the chances of no six are "almost surely"
> >zero, having picked up the phrase from wikipedia.
> >
> >[Thoughts]
> >I understand that the odds of throwing a 6 after n rolls is
> >1-(5/6)^n
> >As n -> infinity, (5/6)^n -> 0
> >Thus I *believe* when n == infinity, (5/6)^n == 0
> >
> >I also understand that in the second statement (consider ...) it is
> >saying that since there are infinite real numbers between 1 and 10,
> >the odds of selecting a particular number is 1/infinity = 0.
> >
> >But surely one of the statements above is wrong since they surely
> >conflict?
Hi,

Thanks for writing to Dr Math. Both statements are correct and they
do not contradict. You are correct that the probability of throwing
at least one 6 among n rolls is 1 - (5/6)^n, and you are correct that
this number approaches 1 as n increases to infinity. But picking a
real number is equivalent to rolling a 10-sided die infinitely many
times, and not just getting 5.0724, but getting a 5 on the first roll,
a 0 on the next roll, a 7 on the next roll, a 2 on the next roll, then
a 4, then a 0, then another 0, and another 0, and another 0, and
continuing to get 0s ever after. The probability of getting at least
one of the first n digits correct is 1 - (9/10)^n, but the probability
of getting *all* of the first n digits correct is 1/10^n, which
approaches zero as n increases to infinity.

[quaddriver]

Are you saying it is possible to rolla 0? Im not playing craps at your house.

[Sean]

> >However, others believe the chances of no six are "almost surely"
> >zero, having picked up the phrase from wikipedia.

That's quite insulting Stoat. It's obvious that I have a better knowledge of maths than you give me credit for. I linked the wikipedia page to 'almost surely' as a courtesy to anyone who may not be aware of its mathematical meaning.

Be careful mate, you seem to think that you're dealing with idiots. That can only backfire on you...



I do have more to add but I want to review your last post again first...

[Sean]

The thread that just won't die.

I had a response from Dr Maths and I append it here in its entirety, along with my original question to them for reference. I think the important comment is "Both statements are correct and they do not contradict" - so the unambiguous "if you have only a finite number of outcomes and you take an infinite number of trials, you will end up getting the outcome you are looking for" does help my argument.

I have asked for further clarification, but I *think* that the statement we (all?) thought contradictory maybe highlights our limit of maths understanding, since now they talk about limits rather than "reaching infinity". Anyway, I shall post back if and when I get an answer.



> [Question]
> >I am having a long argument about the odds of throwing a 6 using a
> >standard dice in infinite time. Your site appears to contradict itself
> >and this has added to the confusion.
> >
> >Here (http://mathforum.org/library/drmath/view/55871.html) you state
> >"if you have only a finite number of outcomes and you take an infinite
> >number of trials, you will end up getting the outcome you are looking
> >for".
> >
> >Here (http://mathforum.org/dr.math/faq/faq.prob.intro.html) you state
> >"Consider the example of picking a random number between 1 and 10 -
> >what is the probability that you'll pick 5.0724? It's zero, but it
> >could happen". Please help
> >
> >[Difficulty]
> >It appears the 2 statements on your web site contradict each other,
> >and this I find confusing.
> >
> >My feeling is that if a dice was rolled an infinite number of times
> >and a 6 did not show, then it was never possible to roll a six.
> >However, others believe the chances of no six are "almost surely"
> >zero, having picked up the phrase from wikipedia.
> >
> >[Thoughts]
> >I understand that the odds of throwing a 6 after n rolls is
> >1-(5/6)^n
> >As n -> infinity, (5/6)^n -> 0
> >Thus I *believe* when n == infinity, (5/6)^n == 0
> >
> >I also understand that in the second statement (consider ...) it is
> >saying that since there are infinite real numbers between 1 and 10,
> >the odds of selecting a particular number is 1/infinity = 0.
> >
> >But surely one of the statements above is wrong since they surely
> >conflict?
Hi,

Thanks for writing to Dr Math. Both statements are correct and they
do not contradict. You are correct that the probability of throwing
at least one 6 among n rolls is 1 - (5/6)^n, and you are correct that
this number approaches 1 as n increases to infinity. But picking a
real number is equivalent to rolling a 10-sided die infinitely many
times, and not just getting 5.0724, but getting a 5 on the first roll,
a 0 on the next roll, a 7 on the next roll, a 2 on the next roll, then
a 4, then a 0, then another 0, and another 0, and another 0, and
continuing to get 0s ever after. The probability of getting at least
one of the first n digits correct is 1 - (9/10)^n, but the probability
of getting *all* of the first n digits correct is 1/10^n, which
approaches zero as n increases to infinity.

I'm afraid that this doesn't really help your stance at all Stoat. Unfortunately the second quote you chose is not the one that contradicts the first. The quotes you needed to ask about were these...

To make a long story short, if you have only a finite number of
outcomes and you take an infinite number of trials, you will end up
getting the outcome you are looking for.

and

Likewise, when dealing with infinities, a probability of 1 doesn't guarantee the event:

Feel free to try again!

[thestoat]

Very sorry Sean - it wasn't meant to be insulting. The "almost surely"" was initially picked up by Andrew. My feeling was (and is) that Andrew (and please correct me if I am wrong, Andrew - I have no wish to insult you either) does not understand this sort of maths that well, and latched on to the "almost surely" in an english sense rather than a mathemetical sense. Anyway, you are of course correct - the question did emphasise the wrong part of the web site statement which is why I needed to go back to them.

I have had another reply from the web site. In reading it, the monkeys are gonna do Shakespeare.

> >Thank you very much for your kind answer. I think I understand your
> >reasoning about the 10 sided dice, and I am happy to know there is no
> >contradiction on your web site. Just one thing evades my understanding
> >though.
> >
> >I can see that a probability of zero does not mean an event is
> >impossible and thus a probability of 1 does not mean an event is
> >guaranteed. This is inferred in the statement "Likewise, when dealing
> >with infinities, a probability of 1 doesn't guarantee the event".
> >
> >However I was under the impression that a probability of 1, by
> >definition, guaranteed the event. This must therefore be false? But my
> >main query is that, given the fact that a probability of 1 does not
> >guarantee the event, in an infinite time, how can the probability of
> >rolling a six be therefore guaranteed? ("if you have only a finite
> >number of outcomes and you take an infinite number of trials, you will
> >end up getting the outcome you are looking for"). (Our original
> >discussion was about monkeys typing Shakespeare, but the dice analogy
> >is presumably just as apt).
> >
> >Thanks again for your time
> >
> >

Hi,

Well, the answer to your question really depends on what you mean by
"guaranteed." You are correct that there is a big difference between
"will happen with probability 1" and "must happen." The simplest
probability spaces are ones where there are only finitely many
possible outcomes (like when rolling a die once, or even finitely many
times), in which case having probability 1 means that it must happen.
But when there are infinitely many possible outcomes, then those two
things are not the same.

Now, in real life, you can bet your life that a zero-probability event
won't happen, in the sense that you can choose a zero-probability
event, then do the experiment, and you won't get that event happening.
(Try flipping a coin until you get heads; you'll find that you
always, eventually, get heads after finitely many flips.) On the
other hand, in many probability spaces, every event has probability
zero. (Pick a random real number between 0 and 1.) In such a case,
the result of any experiment will be a zero-probability event. But
there are so many possible events that might happen (infinitely many
of them, and none more likely than any other) that it would be
impossible to guess the result before the experiment. So does that
mean that you are guaranteed to guess wrong? It is still possible
that you get tails every time you flip a coin; it's just a
zero-probability event.

So one mathematician might say "guaranteed" to mean "probability one"
and another might use the same word to mean that there are no outcomes
where this doesn't happen. But those *are* two different meanings.

[Sean]

No problem mate. Glad we cleared that up.

You are correct that there is a big difference between "will happen with probability 1" and "must happen."

It is still possible that you get tails every time you flip a coin; it's just a zero-probability event.

So one mathematician might say "guaranteed" to mean "probability one" and another might use the same word to mean that there are no outcomes where this doesn't happen. But those *are* two different meanings.

I suppose it's human nature that we will pull out the parts that support our own position.

Taking his words at face value and putting no spin on them it still appears to me that he is on my side. My view is that the probability is actually 0.999999..... but is given as 1 simply because it is so close as to make (virtually) no difference.

My position remains unchanged. While I believe that the die will almost surely (in its mathematical sense of course... ) land on a 6 eventually or the monkeys will type a coherent sentence there is no logical reason to believe that this is an absolute certainty will all doubt removed. As long as there is randomness (a roll of a die, an illiterate monkey) there will always be doubt. By its very definition we cannot control or absolutely predict the outcome of randomness.

[thestoat]

I don't believe he is saying that. It seems to me there are 2 different scenarios. In a scenario with a *finite* number of possibilities then the outcome *will* happen. This is where he states

The simplest probability spaces are ones where there are only finitely many possible outcomes (like when rolling a die once, or even finitely many times), in which case having probability 1 means that it must happen.

The above caters for dice and Shakespearien monkeys which are there are only finitely many possible outcomes. As he says, "it must happen".

So when he says

It is still possible that you get tails every time you flip a coin; it's just a zero-probability event.

this means that it won't happen - zero probability.

He then goes on to the point that if there are infinite number of outcomes (real number between 0 and 1) then the idea of a "zero probability event" has a different meaning.

Thus I still believe, based on the above, that the monkeys are gonna do some great stuff

[Sean]

I'm a little confused as to how you got the meaning "it won't happen" from the words "It is still possible"...

BTW

The simplest probability spaces are ones where there are only finitely many possible outcomes (like when rolling a die once, or even finitely many times), in which case having probability 1 means that it must happen.

You cannot seriously use this to support your argument as it clearly states that it is applicable to a finite number of rolls. We are discussing infinity. Therefore a finite number of rolls does not apply to dice or monkeys (or monkeys rolling dice...).

[thestoat]

I'm a little confused as to how you got the meaning "it won't happen" from the words "It is still possible"...

Because he says

It is still possible ... it's just a zero-probability event.

My understanding of this is that a "zero probability event" means 2 things -
with finite outcomes, it means "it won't happen"
with infinite outcomes it means "it almost surely won't happen"

I arrive at this conclusion due to his final paragraph.

You cannot seriously use this to support your argument as it clearly states that it is applicable to a finite number of rolls. We are discussing infinity. Therefore a finite number of rolls does not apply to dice or monkeys (or monkeys rolling dice...).

Well if it "must happen" in a finite number of rolls then surely it "must happen" in an infinite number of rolls. You are talking about the number of rolls (or tries). His distinction talks about the number of possible outcomes determining the ultimate probability -

But when there are infinitely many possible outcomes, then those two things are not the same.

What he is saying is that there is a difference between measuring probabilities for finite outcomes (dice, monkeys) and infinite outcomes (real numbers).