Oh dear god. Math nerds have taken over this thread. We are doomed.

RvW, that is exactly why I said it was wrong...

The point was that we don't compare "luck" on specific permutations, as it was mentioned every one has the same probability. The question at hand is "how probable it is to get the right coinflip 5 of 12 times compared to getting the right coinflip 0 of 12 times".

That is pretty much the only possible mathematic model we could operate on, and in that model for example (H,T,T,H,T,H,T,H,H,H) is indistinguishable from (T,T,T,T,H,H,H,H,H,H,). When we compare how "lucky" or "unlucky" we were we should not distinguish turns from each other, because on every one of them (bar OP cards) the coinflip is the same. In mathematics it is described as binomial distribution (related to Bernoulli) and is the most basic tool used when stydying cases like that:

http://en.wikipedia.org/wiki/Binomial_distribution

Please try to use the probability mass function to calculate both occurances (Red's example and yours), you should quite clearly see the difference.

Ah, then I see where we misunderstood each other; I actually did mean the exact order (not merely number) of occurrences. Hence, "H,T,T,H,T,H,T,H,H,H" is distinct from "T,T,T,T,H,H,H,H,H,H" (maybe I should've written it as "P(F1=H, F2=T, F3=T, F4=H, F5=T, F6=H, F7=H, F8=H, F9=H, F10=H)" instead?).

BTW, I'm slightly disappointed nobody mentioned the prime numbers...

Yes I understand what you meant, but it just made no sense at all to view first orders in specific permutations. If you want to analyse/measure being "lucky" or "unlucky" which are just simple words for "deviation from expected value" then we cannot distuinguish turns between themselves...

Also since you requested prime numbers and I am very much into number theory...

Here you go, a closed formula for n-th prime number ;)

I don't think you do understand what he meant. What he was saying was something along the lines of "it all depends on what you perceive as luck" (which is quite close to what you are trying to correct him to). To show this, he gave a random example that was just as likely, yet nobody would care about at all - showing that the human mind only finds things unlikely in special cases ("unlucky" cases, cases with a pattern to them, homogenous cases).

So basically, you are saying the exact same thing (except for the part you claim he's wrong and he claims he is right).

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I really have no strength to explain the basics of buldin probability models...

We want to observe who gets first order right? Since it's a coinflip we expect both players to come out on top half of the time (bar OP cards). So when do we say one player was unlucky? When the outcome deviates far from the expected outcome... With perspective he provided there was no luck at all, with his perspective no outcome would be unlucky or lucky. So basically what I am saying is if we are using words "lucky", "unlucky" we mean something that does not have to be described with great detail but basically means an anomally, a deviation from expected outcome. Of course here we are talking globally, about first orders through the whole game so it is quite clear how we should implement that into our model if we want to analyse it. Of course if we go into the game deeper we might call lucky many other things when we compare it to the outcome of the whole game. Anyway in case of that global approach to first orders I think the case is closed...

Jasper, actually me and szeweningen do indeed agree. I was just talking more in general, while he was still looking at "luck".
From his point of view the order of events is irrelevant (we just want to know how many times a given player got first turn, "lucky"), while from my point of view (talking more in general, semi-ignoring how the discussion got started) the order of events is relevant. That's a huge difference.

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szeweningen, regarding that (black on dark-grey, gotta love transparent backgrounds :p ) formula for primes, the part after the multiplication looks very weird...
The innermost part is "1 + floor(k/i) - k/i" (with k and i both strictly positive). Unless I'm mistaken "floor(k/i) - k/i" will always be somewhere between 0 (inclusive, if i divides k) and -1 (exclusive). That puts "1 + floor(k/i) - k/i" in the range (0, 1], after flooring it will be either 0 or 1.
Because the final iteration has i=k there will be at least one case where it's a 1 (good, wouldn't want to divide by zero!), so the total sum is at least 1. Then we take the reciprocal and floor again... giving either 1 (if all other terms came out 0) or 0 (if at least one other term came out 1).
Since terms are 1 when i divides k..., does that second part simply say "if prime (k) return 0 else return 1"...!? While I do understand the appeal of closed formulae, my (computer science) heart weeps at the ridiculously inefficient (not to mention horribly roundabout and utterly unreadable) way of expressing it. (For some reason, I have to think about GEB now. :) )
Still, can you link the paper (??) where you found that beast? I'm very curious about the explanation the "inventor" him/herself gives. So far, the "best" result I knew about primes is that there are approximately n/ln(n) primes smaller than n; I "remember" (probably incorrectly) having been taught that such a formula as you just gave is supposed to be impossible...

Please oh please tell me you guys get paid to do this. Nobody talks about math this much, or takes it this seriously, who doesn't get paid.

Hehe, well, that is a tricky formula. Of coourse there is no useful formula for n-th prime number that'd help in computation of them, all formulas like that are based upon some tricks that basically do what you said (if p prime return sth, if p prime return sth). There are many more examples, probably the easiest one to see comes from the Wilson's theorem:
For every n


The formula I gave has a longer and a bit more convoluted explanation, but follows similar lines (try the last one). Anyway you remember correctly, there is no useful formula, the ones existing have no real mathematical or practical value.

And sorry for black background, normally LaTeX works on standard white, but copying image does not include background.

I really hate arguments that end up in "oh, we actually agreed all along".

Any way, I'm not going to argue along since I'm sure I'll end up in the same situation, I'll just leave you with some general comments. One is that luck is perceived to be both a poor excuse for pointing out you're the winner AND for being the loser, though I admit that's not always the case. The latter is quite common. I myself absolutely hate that any attack of 13 or lower against neutrals of 2 with a luck percentage of 75% has a chance to fail. Most games last too short to have certain luck or bad luck even out. This means particular games with 75% luck can end up having a lucky winner instead of a skilled winner. But this is not really a problem. You play Warlight knowing there's a huge luck factor involved. Only if you set luck percentage to 0%, give everyone more starting positions, etc, can you almost entirely remove luck from the game and have it always be a skilled win. Unfortunately, only members can do this.

Another comment is that I don't think it's necessarily bad you never get first order. Usually you only want first order to quickly take out an army trying to gain access to your bonuses, taking a territory to stop an enemy's army in their tracks, or to transfer a big army in defense against your opponent.

szeweningen:
Just threw that first formula of yours into Mathematica. I'm getting

{
  {   0.000, 1}
  {   0.000, 1}
  {   0.000, 1}
  {   0.000, 1}
  {   0.016, 1}
  {   0.281, 1}
  {   1.809, 1}
  {  15.226, 1}
  { 119.902, 1}
  {1160.1  , 1}
}

when I use Timing[] to tabulate the values of n from 1 to 10. The first is the number of seconds it took for that row (20 minutes on the last one...), the second is, allegedly, a prime number...!? (Where it says "0" it originally said "1.46367*10^-17", meh.)

That second formula looks incomplete (did you cut of a quantor at the front?); the exponent (with superfluous brackets around the second half), cancels out to "1" so it can be removed, after which the "2*" and "/2" cancel out as well leaving us with just "2n+1". Assuming n to be a natural number, that claims that (among many others) 9 is prime... (Allowing integers, rational, ... obviously breaks even worse. Restricting n to primes doesn't work either, since 2*7+1 = 15.)

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@Darkruler, second paragraph: ah yes, good point; the existence of the Order Delay Card nicely illustrates this.

My head asplode.

^^ All this right here is why I'll never be the best player. Math isn't strong with this one.

RvW, this [] means in programming language \floor{}, not a regular bracket

The second formula I believe is, as I remember, a nice way of using Wilson's theorem to make a tricky formula, which can be described as:

Let f be our function $f:\mathbb{N}\rightarrow \mathbb{N}$

$(\forall_{p\in \mathbb{P}} \ f(p)=p\in \mathbb{P})\wedge(\forall_{n\notin \mathbb{P}} \ f(n)=2\in \mathbb{P})$

I think I encountered that problem in 10th grade or so, it is pretty funny how you can make a closed formula using Wilson's theorem (without it it'd be pretty hard to give a closed formula if it was to be described as I did). Anyway I used LaTeX language, but if I remember correctly, mathematica uses the same symbol description.

The first formula uses straightforward floors and an absolute value to be exact, the second one uses [] as \floor{}.

The first formula uses straightforward floors and an absolute value to be exact, the second one uses [] as \floor{}.

Okay, then I did enter the first one correctly (wasn't sure about the absolute value bars; thought maybe those meant something different).
That second one (now I know it's supposed to be a floor) makes my head hurt trying to simplify it. :s

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@noobschool:
Don't worry, we've gotten rather off-topic; the whole prime number business is unrelated to WL. (But it's indeed definitely possible to use other areas of mathematics to your advantage!)

RvW, which ones?
Well I know about counting, but besides that?

Actually a few:

- probability methods in game theory (that's a huge topic, maybe before I retire I'll write sth about that)
- income stream as Markov chains (or a simpler version of that to find optimal expansion strategies)
- optimalisation in terms of variables, incresing expected value while keeping variation under control (for example knowing you opponents income in 1vs1 and keeping 2 fronts with him, predicting full deployment on borders you want to use 2 attacks, which can be represented by X(a)+Y(b) where a+b is your income)

Those are of the top of my hat and I believe with those tools we could build a comprehensive (though complicated) method of analysing games in those terms. Those are of course theoretical mathematical methods and there are tons of related stuff in the field of IT, for example a well-built neuron-web would provide an incredibly strong AI granted it plays on one setting and gets to play only strongest opponents.