In the beginning ...

[MajGenl.Meade]

You chickens!

[Sean]

Ummm... cluck?

[Rick]

I'm still having a problem with the rolling of the di.

The chances of rolling the di once and coming up with 6 is 1/6 the chances of rolling it seven times and coming up with six every time is not 1/6...

[Big RR]

No, but even if you roll 5 sixes before, the chance that it will be six on the next roll is still one out of six, as the result of any roll is independent of any previous roll.

[Scooter]

The probability of rolling 7 sixes is 1/6 x 1/6 x 1/6 x 1/6 x /1/6 x 1/6 1/6 = 1/279936

Which is the same probability of rolling 2,4,5,1,6,3,5
Which is the same probability of rolling 5,1,4,6,3,2,4

Whatever outcome occurs from rolling a die 7 times, the probability of that outcome is always 1/279936.

Edited because the examples I used were of only six rolls each

[thestoat]

Scooter and Big RR are spot on ...

[Rick]

Scooter and Big RR are spot on ...

Cool, you pay me based on Scooters odds I pay you based on BigRRs...

[thestoat]

Scooter and Big RR are spot on ...

Cool, you pay me based on Scooters odds I pay you based on BigRRs...

Sounds about right - the odds of me paying you are roughly in line with Scooters odds

[thestoat]

I'm still having a problem with the rolling of the di.

The chances of rolling the di once and coming up with 6 is 1/6 the chances of rolling it seven times and coming up with six every time is not 1/6...

In that specific case it is Scooters calculations that are relevant.

For a normal dice, no matter how many times you roll, no matter what numbers you roll, the chances of the next roll being a 6 is 1 in 6 as Big RR states.

[Big RR]

But if you ask before you ever roll "what are the chances that I will roll six 7 times" (or any other specific sequence of numbers between 1 and 6), then the odds are the ones Scooter posted.

[Andrew D]

This is key:
"While there is no difference between almost surely and surely"

The difference is that I believe you are "reasoning about infinity by imagining a vast but finite number"

By conveniently omitting a portion of the sentence, you have attempted to change its meaning. That (rather underhanded) attempt fails. What the sentence actually says is:

While there is no difference between almost surely and surely (that is, entirely certain to happen) in many basic probability experiments, the distinction is important in more complex cases relating to some sort of infinity.

We are dealing with infinite numbers, not with basic probability experiments. Therefore, the distinction between almost surely and surely is important.

As to which of us is "reasoning about infinity by imagining a vast but finite number," the answer is self-evident. One of us wrote:

... 9999999999999999999999999 millennia (which we could approximate to an infinite time) ....

What clearer example of "reasoning about infinity by imagining a vast but finite number" can there be than attempting to "approximate [a finite number] to an infinite [number]"?

[Andrew D]

5 * 0 = 0.
8 * 0 = 0.

This does *not* mean 5 = 8

Similarly

1 * ∞ = ∞
60 * ∞ = ∞

This does *not* mean 1 minute = 1 hour

The examples are not at all parallel. We can see this clearly by expressing the ideas in a natural human language.

If I have a basket containing no apples, then I have no apples. If I add to that another basket containing no apples, so that I have two baskets each containing no apples, then I still have no apples. If I multiply that by a billion, so that I have two billion baskets each containing no apples, then I still have no apples.

These:
0 (which is the same as 1 x 0) = 0
0 + 0 (which is the same as 2 x 0) = 0
2,000,000,000 x 0 = 0
make perfect sense, because "zero" means "nothing". If I add nothing to nothing, I end up with nothing; and if I do that two billion times, I still have nothing.

So there is the imparallelism: Whatever ∞ means, it self-evidently does not mean "nothing".

● If I have a basket (basket A) containing ∞ apples, that basket necessarily contains a number of apples greater than zero.
● If I also have another basket (basket B) also containing ∞ apples, that basket also necessarily contains a number of apples greater than zero.
● If the number of apples in each basket is the same as the number of apples in both baskets combined, then when I add the contents of basket A to the contents of basket B, I have added nothing to the contents of basket B.
● If by adding the contents of basket A to the contents of basket B, I have added nothing to the contents of basket B, then the contents of basket A are nothing.
● But the contents of basket A are not nothing; basket A contains ∞ apples, and ∞ ≠ 0.

This is why people claiming that the number of minutes in an infinite number of hours is the same as that infinite number of hours, that 60 x ∞ = ∞ (when ∞ infinity is taken in both instances to be the same infinite number), etc. always resort to the artificial "language" of mathematics: When expressed in natural human language, the propositions simply make no sense. And that, as I posted before, proves that some error(s) lie(s) in the mathematics.

[Andrew D]

The non-believer (admit it or not) must argue that time has no beginning – that the universe (or call it what you will) has infinitely existed in some form or another – a singularity, an eternal oscillation from one state to another. It ignores the inconvenient fact that in infinite time (and they mean an infinite regression into the “past”), all that could possibly happen must have happened. In which case we would not be here at all by now. They want all this time so that they can deny God and guess that mankind emerges by accident, but want to avoid the inescapable conclusion that entropy would already have had to occur – for that is what a true “infinity” of time must produce.

I have yet to see any demonstration that what you call an "inconvenient fact" is a fact at all, inconvenient or otherwise. That is what I am asking for.

What I believe or do not believe has no bearing on that question. Your quoted argument depends on what you call a fact's actually being a fact. Look what becomes of it otherwise:

The non-believer (admit it or not) must argue that time has no beginning – that the universe (or call it what you will) has infinitely existed in some form or another – a singularity, an eternal oscillation from one state to another. It ignores the inconvenient fact possibility that in infinite time (and they mean an infinite regression into the “past”), all that could possibly happen must may or may not have happened. In which case we would not be here at all by now, unless we would. They want all this time so that they can deny God and guess that mankind emerges by accident, but want to avoid the inescapable conclusion that entropy would or would not already have had to occur – for that is what a true “infinity” of time must produce.

Rather less persuasive, don't you think?

[MajGenl.Meade]

I have already provided the argument which you merely contradict.

An impossible thing is that which can never happen
A possible thing is that which can happen
A possible thing can only happen in time and there is no time greater than "infinite time"
If a thing does not happen given infinite time in which to happen, then it is an impossible thing
Therefore, in infinite time all impossible things will not happen and all possible things will happen
An infinite regress of time includes all time that can happen and is neither larger nor smaller than any other infinite
Therefore in an infinite regress of time, all possible things will have happened
For a thing that cannot happen in infinite time is an impossible thing
Therefore there is not an actual infinite regress of time
Therefore the universe had a beginning and has not eternally existed

Inelegant I'm sure
Cheers
Peter

[loCAtek]

It's called the 'Hero Myth' by Joesph Campbell (you could include Luke Sykwalker in there too) The ordinary doing extraordinary, ya could call it spiritual...

[thestoat]

By conveniently omitting a portion of the sentence, you have attempted to change its meaning. That (rather underhanded) attempt fails

Not underhand - merely trying to keep things simple.

the distinction is important in more complex cases relating to some sort of infinity.

I read that to mean discussing other sortS of infinity (I have read there are several), not the simple infinity we are discussing. My mistake.

As to which of us is "reasoning about infinity by imagining a vast but finite number," the answer is self-evident. One of us wrote:

thestoat wrote:
... 9999999999999999999999999 millennia (which we could approximate to an infinite time) ....


What clearer example of "reasoning about infinity by imagining a vast but finite number" can there be than attempting to "approximate [a finite number] to an infinite [number]"?

Now you are quoting out of context - though I don't believe you are being underhand. Approximations exist all the time in maths and physics and this is what I was referring to (read on where I talk about parallel lines of light from the sun). That does not apply in this case where we discussing actual infinity. Very, very different.

[Sean]

Therefore, in infinite time all impossible things will not happen and all possible things will happen

This for me is the fallacy. I agree that impossible things will not happen (by their very nature) but there is no reason to say that all possible things will definitely happen. Of course I like to think that I am open-minded and would welcome any evidence to prove me wrong.

[thestoat]

This is why people claiming that the number of minutes in an infinite number of hours is the same as that infinite number of hours, that 60 x ∞ = ∞ (when ∞ infinity is taken in both instances to be the same infinite number), etc. always resort to the artificial "language" of mathematics: When expressed in natural human language, the propositions simply make no sense. And that, as I posted before, proves that some error(s) lie(s) in the mathematics.

I introduced the zero against your assertion

The proposition that the number of minutes in an infinite number of hours is equal to that infinite number of hours leads to the conclusion that a minute is equal to an hour.

because with that assertion you could substitute zero for infinity. Again, you are trying to treat infinity as a real number - which it is not.

You are trying to translate from imaginary to real numbers and this cannot be done in such a simplistic way. If you re-read the posts you will see

a . ∞ = ∞

If you will not accept that then you are not accepting the wisdom of the vast intellect that is maths and there therefore won't anything I can say to change your mind. It is a fundamental tenant of maths.

Thinking about the baskets

A = ∞
B = ∞
If you add the contents of A to B
A = ∞ - ∞
B = ∞ + ∞

Now, if you look at the equations I linked, you will see there is no equation for either of the above. You cannot add or subtract infinities - that is not defined and this is where your maths breaks down.


Finally, I have found some help with the original assertion about the monkeys and typewriters. The result is:

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.

You can read it here: http://mathforum.org/library/drmath/view/55871.html

As I said previously, you are treating infinity like a really large number, and that is wrong. Infinity can be treated like a really large number when using approximations. But not when it needs to be infinity.

[Andrew D]

Here's the problem (again):

If a thing does not happen given infinite time in which to happen, then it is an impossible thing

That is a leap, not a logical progression. "Impossible" means that it cannot happen. But you keep substituting "does not" for "cannot".

How do you justify that leap?

[Andrew D]

It is a fundamental tenant of maths.

* * *

... this is where your maths breaks down.

No. The breakdown is in the mathematics by which people claim to have shown that the number of minutes in an infinite number of hours is equal to that infinite number of hours.

The problem is simple to state: The conclusion is either patent falsehood or babbling nonsense (or both).

When someone can explain in a natural human language (which, on this form, means in English) how it is that the number of minutes in an infinite number of hours is equal to that infinite number of hours without a minute's being equal to an hour, then we'll be getting somewhere.

But as long as the refuge continues to be that "'the maths' say so," we will get nowhere.

Yes, apparently, the maths say so. And what they say is, when expressed in natural human language, either patent falsehood or babbling nonsense (or both).

Therefore, something is wrong. And the error must lie in the mathematics.

Why (in English) is the statement "the number of minutes in an infinite number of hours is an infinite number sixty times greater than the infinite number of hours" wrong?

We have no difficulty saying that the number of (fluid) ounces that represents a quantity of liquid is thirty-two times the number of quarts that represents the same quantity of liquid. And we have no difficulty saying that that is true whether one has one quart (thirty-two ounces) or a billion quarts (thirty-two billion ounces) or a googleplex quarts (thirty-two googleplex ounces) of liquid.

So why (in English) should we throw that obvious equasion of thirty-two ounces to a quart overboard when we deal with infinite numbers? Why should the infinite number of ounces in an infinite number of quarts suddenly -- presto! abracadabra! -- become the same as the infinite number of quarts?

And what is the problem -- in English -- in saying that the infinite number of minutes in an infinite number of hours is an infinite number sixty times greater than the infinite number of hours? We all know that the number of minutes in a finite number of hours is a finite number sixty times greater than the finite number of hours, so why is that not the case when dealing with infinite numbers?

(As to confounding finite and infinite numbers, you were the one who wrote that we can "approximate" a finite number to an infinite number. If there is a problem, it is yours, not mine.)