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[neon snake]

A few points.

Firstly, I'd like to echo a couple of statements.
1) This has been for the most part, entirely enjoyable. I've learned some things, remembered a lot that I thought I'd forgotten, and generally thought more about maths than I have since finishing my degree.
2) Yes, please let's keep this friendly, on all sides. The few days between Christmas and New Year were NOT enjoyable, as we all got involved in name-calling, and being generally unpleasant. For my part, I apologise if any of my posts can be included in the general unpleasantness.
3) thebruce, I'm quite willing to shake your virtual hand. I'm also quite happy to continue, if we can keep the tone at the level it is currently at.
4) I think the 'E numbers' thread may indeed enter PPC legend. Good on us. Lets hope when the game starts that we apply as much though to the puzzles as we have here.

I'll end this post here, as I want it to stand seperate from the next one...

[neon snake]

Right, let's get on with it.

I think I've got (more-or-less) my bullet points.

Some of us assert the following (please let me know asap if I've worded any of these incorrectly, so I can edit).

1) If a string is both random and infinitely long, then the chance of any given finite string appearing is exactly 100%.

2) 0.9recurring = 1.

3) An infinite sequence can equal a finite sum.

4) In the number e (infinite, but non-random, as it can be calculated), the chance of a given finite string appearing is either 0% or 100%. It can't be both at the same time.


Note that these statements are not necessarily connected, neither do they follow on from each other.


Some of us ( ) refute these statements.

Correct?

[neon snake]

Now, I'll leave 2 and 3 alone. Whilst I agree with them, my maths isn't good enough to prove them. There are others here with higher ability than myself.

4 seems hardly worth bothering with. If we know the values in a string of digits (infinite of otherwise), then we can obviously determine whether a second, finite string appears or not. If it does, the 'chance' is 100%. If it doesn't, it's 0%.

I may have slipped up there, but I don't think so. SilentAvenger, you seem to have the knowledge on this one, so any thoughts from you on my statement above would be appreciated!

That leaves me with 1.

I'll start again, and see where we get.

My assertion is:

'If a numerical string is infinitely long, is truly random and contains all of the digits from 0 to 9, with each having a chance of being the next digit in the string,

then

the chance of any finite string appearing is exactly 100%.'

I think the statement covers all my bases, and states the assumptions clearly.

Right. Firstly, the statement assumes that infinite trials/tests/whatever are possible. If disputing my statement, I ask that you accept this. Please do not introduce arguments along the lines of:
'It's impossible to perform an infinite amount of trials, you'd die'
Let's stick to trying to prove/disprove the portion after the 'then'!

As pointed out by SilentAvenger, this statement can be proved using Kolmogorov's zero-one law (which I had never heard called that, interestingly).

Definition here- http://en.wikipedia.org/wiki/Kolmogorov%27s_zero-one_law

Is the 'event' (a set finite sequence being found) a tail event?

Yes, as it is probabilistically independent. IE it does not rely on anything that has gone before it (similar to the coin-toss example included in the definition).

Is there a chance that the 'event' might happen?

Yes, since all numbers each have a chance of appearing, then it is actually possible.

Therefore, since it is possible, and it is a tail event, then the probability of the 'event' eventually occuring must be 1 (or 100%).



I believe that to be robust.

[SilentAvenger]

1. thebruce, 0.3~ is not irrational, irrational = cannot be written as the quotient of two whole numbers, not cannot be displayed in decimal form.

2. neon, about e, I'm not even sure myself. It *is* a calculated number after all, and that might screw around with some stuff, but yea, the chance, because e is infinite, has to be either 0 or 1, I'm just not sure if you can determine for a given string which one it is, without going and looking for it.


*agrees to virtual hand shaking*
*hopes this will enter PPC history along with the postcard code*

notice: HISTORY, as in, stuff not happening right now.

oh, and on a side note, we were forced to prove in class today the sum (a1/1-q) of infinte regressive geometric series (those that get smaller all the time ->0), in a twist of irony

[Chris K]

We don't know all the digits of e so I think it must be impossible to know if any given string will appear in it.

I guess that e will be fairly important in the game so here are some fun facts about it:

- It is named after the brilliant Swiss mathmatician Leonhard Euler
- It is transcendental
- It can be calculated with the infinite series:
2 + 1/2! + 1/3! + 1/4! + 1/5! ....
Or the limit:
(1 + 1 / x ) ^ x
- e ^ x differentiates onto itself
- The area under the graph y = 1 / x between x = 1 and x = e ^ n, is n, For any value of n.
- It is the base of the natural logarithm, ln

[Centipede]

It is equal to the mass of the object times teh speed of light squared.


....no, wait, wrong 'e'

Dammit why do they need to keep reusing letters and symbols for different things? Why couldn't Einstein's 'E' be like a lightning bolt or something?

[Rei]

Ok I gave up reading this properly a while ago (i normally read it at 1am, doesnt go down well)

I talked to my maths teacher today (masters maths) who descided that the probability of any combination/permutation of numbers turning up in something like e is 0, not 1.

[SilentAvenger]

Rei, consider the finite sequence "1". It obviously appears in e. -> Teacher is wrong.

Centipede, Euler found that out. Thats why he called degrees of infinity with the hebrew letter aleph

[Olorin]

[Olorin]

[Rei]

Hmm it was at 9:30am when i talked to her... and i think i've forgotten bits of it... anyway she disagrees with you.

mooo! (i give up again, dont respond directly to me please -_-)

[neon snake]

I think what teacher probably mean't was finding all sequences, rather than any.

[thebruce]

Ok, I'll do my best to reply to your points, since it seems the discussion still wants to continue, by request and comments

[Chris K]

Good to see you're still flat out denying that 0.9~ = 1.

Can you provide any links that say you can't work in decimal with recurring numbers?

Is there anything wrong with this:

0.2~ * 3 = 0.6~

?

[neon snake]