Against all odds!

[wattro]

Originally posted by Axxaer


Sorry, but if you're going to take this track, then I'll have to point out that as the theoretical concept of infinity is, well, infinite, and thus has no numerical value, any number divided by infinity will result in a zero probability because a sample space cannot (to human knowledge of the concepts of infinity and sample spaces) ever be infinite, and as Kataphraktoi rightly pointed out, any outcome which has a factor of 0 outcomes will also have a 0 probability of being a future outcome. Remember that mathematical probability is not the absolute probability of the possible chance of something happening given all known forces, but is rather a prediction based on current/prior knowledge, and will thus rarely equate to the absolute probability. (E.g. Take a regular perfectly symmetrical coin. You currently have an absolute probability of 0.5 of landing on either side, but a mathematical probability of 0 of landing on either side. Flip it once. You still have the same absolute probability but now have a 1.0 mathematical probability of landing on one side and a 0.0 mathematical probability of landing on the 'flipside'. Each time you flip it, absolute probability says it is just as likely to land on either side, although mathematical probability tends to say one will be more likely than the other based on the gathered sample space. As the sample space grows, it is absolutely and mathematically probable that the mathematical probability will converge towards the absolute probability, although based on pure mathematical probability from the current sample space, this is generally not the predicted outcome. (I wonder why Douglas Adams didn't include that explanation in the Hitchhiker's Guide's explanation of why the universe's population is zero )

So to you people who disagreed with Kataphraktoi's (mathematically) correct statement that:

I still disagree...

I probably should have specified limits, as that is what we are really talking about here. I didn't for sake of simplicity, but I could/should have said 78 trillion gazillion bajillion (which might be a plausible estimate of the number of attemtps required to re-order the deck). Therefore I am now creating a mathematical and absolute probability.

I also don't know what the heck you are talking about with mathematical probability. It does not necessarily rely on current/prior knowledge, it relies on number of 'favorable outcomes' out of 'total outcomes'. If we measure a coin flip: one outcome is heads, one outcome is tails, there are two total outcomes. Either way you look at it, the probability that the coin will yield heads is 1/2, tails is 1/2. Any result is 1/1 (according to your theory, you will not get a result since it has not occured before, thus 0/1 - what?!?!) - this case above is a demonstration of both mathematical and absolute probability.

Absolute probability is only looking at the current sample (your wife has 8 kids, she has one more, the absolute probability it will be a boy is 50%), in which case, I agree, a deck of cards will never re-order. But the mathematical probably is, yes, they will eventually, and inevitably, re-order.

[Blake]

this "Mathematical probability" seems to be some kind of black box probability. Like a black box that can give different results.

First time it pops out HEADS. The best guess is a 100% chance of it producing HEADS. I can't remember enough of advanced Statistics to know if it spitting out HEADS a second time increases the confidence of it producing HEADS.
Statistically, is there are a difference:
Spitting out HEADS once. 1/1
Spitting out HEADS one-thousand times. 1000/1000.

Anyway, you don't know what the blackbox is going to spit out next. It could be HEADS agian, it could be TAILS, it might be FEET, HANDS, ARMS, LEGS and TORSO.

I'm not sure what this has to do with combat odds or evolution though since they aren't black box systems.

Except in the "silly" case, of like flipping a dice:
There's a chance it'll be HEADS or TAILS.
There's a smaller chance it'll be SIDE.
There's a miniscule chance it'll be ALIEN INVASION FLEET UNCLOAKS IN EARTH ORBIT AND BLASTS THE COIN INTO MONOATOMIC VAPOR.
In this sense the universe is a black box of sorts since we can't fully understand or predict what is going to happen, but that doesn't mean we shouldn't try, using sensible statistically methods.

[Axxaer]

I haven't had a good mathematical argument like this for years... I don't even know if I'm right I just like talking about it

[wattro]

Originally posted by Blake
this "Mathematical probability" seems to be some kind of black box probability. Like a black box that can give different results.

First time it pops out HEADS. The best guess is a 100% chance of it producing HEADS. I can't remember enough of advanced Statistics to know if it spitting out HEADS a second time increases the confidence of it producing HEADS.
Statistically, is there are a difference:
Spitting out HEADS once. 1/1
Spitting out HEADS one-thousand times. 1000/1000.

Anyway, you don't know what the blackbox is going to spit out next. It could be HEADS agian, it could be TAILS, it might be FEET, HANDS, ARMS, LEGS and TORSO.

I'm not sure what this has to do with combat odds or evolution though since they aren't black box systems.

Except in the "silly" case, of like flipping a dice:
There's a chance it'll be HEADS or TAILS.
There's a smaller chance it'll be SIDE.
There's a miniscule chance it'll be ALIEN INVASION FLEET UNCLOAKS IN EARTH ORBIT AND BLASTS THE COIN INTO MONOATOMIC VAPOR.
In this sense the universe is a black box of sorts since we can't fully understand or predict what is going to happen, but that doesn't mean we shouldn't try, using sensible statistically methods.

Wow... just wow

The coin analogy was used because it (generally) has only two results similar to combat in civ - you either hit (heads) or you get hit (tails).

The chance of getting heads 1000 times in a row is not 1000/1000 (i know you didn't mean it that way) and for the math pundits, it is actually (1/2)^1000.

The best guess is not to think heads just because heads already appeared... unless maybe you are flipping a 2-sided coin. You know there are two probable outcomes and unless you think some external force is forcing heads to appear each and every time, well, at max... you'll be 50% wrong Assume that over a large sample size, the numbers should work out nearer to 50%, not the other way around. Of course not all samples will generate 50%.

The problem people are talking about is losing battles where they are 80% odds to win. That also means that each round of combat, the AI unit has a 20% chance of winning the round against the Player unit.

Assume the battle takes 4 hits for either unit to die. To do this 4 times in a row for the AI is extremely unlikely - (1/5)^4 = 0.16%. It is much more likely for the Player unit, as there is a (4/5)^4 = 40.96% chance that the Player unit will make the first 4 hits.

The numbers say it is probably unrealistic for what people here are reporting and/or experiencing.

/on an unrelated note, I personally think stacked combat isn't built right

[uberloz]

Originally posted by wattro
The coin analogy was used because it (generally) has only two results similar to combat in civ - you either hit (heads) or you get hit (tails).

Coin flipping is not a 50/50 prospect.

Having a modern coin land on its edge is unlikely but still probable.

I think the odds of flipping a modern coin and having the coin land standing on its edge ( correct me please if I am wrong ) are 1 billion to 1.

I just think that the 'extreme odds breakers' were entirely too prevalent in the game I mentioned.

All the examples I gave were from the same game and the same 30 odd rounds of combat during the same war against the same enemy ai.

To have 'superhero units' emerge under the normal circumstances of combat in Civs 4 should be statisticly possible.

I have no problems with that.

What I have issue with, is a proliferation of statisticly improbable results replacing legitimate gameplay under the guise of... 'It can happen, so it does happen!'

Instead of designing an ai that can fight, programming leans towards an abuse of probabilities in an attempt to make the computer appear more worthy.

I want to thank all the mathemagicians and statisticians that contributed to the postings in this thread. I read them all and I never once found them to be 'Off Topic'. Especially since my thread is about probabilities and improbabilities.

Thank you all for donating your time and contributions here.

[Mergle]

Originally posted by wattro


The problem people are talking about is losing battles where they are 80% odds to win. That also means that each round of combat, the AI unit has a 20% chance of winning the round against the Player unit.

Assume the battle takes 4 hits for either unit to die. To do this 4 times in a row for the AI is extremely unlikely - (1/5)^4 = 0.16%. It is much more likely for the Player unit, as there is a (4/5)^4 = 40.96% chance that the Player unit will make the first 4 hits.

The numbers say it is probably unrealistic for what people here are reporting and/or experiencing.

/on an unrelated note, I personally think stacked combat isn't built right

Wow, back on topic!
I think you've misunderstood what the game tells you using the 1.52 patch (thanks to DeepO). It tells you the overall chance of winning the whole combat, not one round. So if it says 80%, you will (on average) lose 1 in 5 times. In any decent war, that will happen several times.

[Mergle]

Originally posted by uberloz


What I have issue with, is a proliferation of statisticly improbable results replacing legitimate gameplay under the guise of 'It can happen, so it does!'

Instead of designing an ai that can fight, programming leans towards an abuse of probabilities in an attempt to make the computer appear more worthy.

You seem to be accusing the programmers of deliberately messing with the RNG - effectively, of making the odds screen lie to you. I really don't think that's happening. Firaxis has a long history of troubles with RNG (see the random map problem) but accusing them of deliberate skewing is I think unfounded.

It's a wellknown psychological problem that people only notice unusual happenings and forget all the times things happen as expected.

[Bhruic]

Originally posted by uberloz
Coin flipping is not a 50/50 prospect.

Coin flipping is 50/50, as long as you are rounding to integers. The chance of getting something other than heads/tails is, statistically, insignificant. Doesn't mean it can't happen, but it does mean that you don't need to consider it for what one might call "normal" scenarios.

Having a modern coin land on its edge is unlikely but still probable.

I hope you meant 'possible'. It certainly is not 'probable' in any sense of the word.

I just think that the 'extreme odds breakers' were entirely too prevalent in the game I mentioned.

It's impossible to say. We only have your word on what happened. I'm not implying that you are lying, but I am saying that your recollection may be off. Or you may not be aware of other factors that were influencing the combat. Or other issues. The point is, without a repeatable test that everyone can look at, it's too difficult to say if you experienced an excessive amount of 'extreme odds breakers'.

What I have issue with, is a proliferation of statisticly improbable results replacing legitimate gameplay under the guise of 'It can happen, so it does!'

Instead of designing an ai that can fight, programming leans towards an abuse of probabilities in an attempt to make the computer appear more worthy.

That would be a poor option in any case. The degree to which random numbers favour the AI vs the Player is equal to the degree to which random numbers favour the Player vs the AI (which is to say not at all). The random number generator doesn't know anything about the situation when it generates a random number, it just makes one. The application of that number does not depend on what the number is. Or, to put it another way, you are just as likely to see a 'fluke' win for the AI as you are the Player, assuming all other factors are equal. Of course, the AI tends to get combat bonuses that mean that it may appear to get 'fluke' wins more often - but that's just a factor of the combat bonuses, which isn't the relevant point.

Bh

[uberloz]

?

[Enigma_Nova]

When rounding to integers, Probability is either 0 or 1. =)
Oh, and I think it's time to break out the lock.

[uberloz]

Originally posted by Mergle
You seem to be accusing the programmers of deliberately messing with the RNG

?

Originally posted by Bhruic Of course, the AI tends to get combat bonuses that mean that it may appear to get 'fluke' wins more often

?

Is it possible that I experienced a 'statistically improbable game'?

"Yes, as a scientist I must concede that, I must volunteer that" ( did anyone else recognize Jodi Foster's lines from the movie 'Contact'?

I have no 'evidence' so feel free to dismiss this entire thread.

[Enigma_Nova]

When posting a thread, would you please be so kind as to include more substantial comments as '?'.
I know you may be confused, but if you're just going to say that, you might as well say +1 - so that instead of being a spamming idiot, we just think you're spamming.
Oh, +1.
The thread will shortly be disregarded and/or locked.

[uberloz]

Originally posted by Enigma_Nova
When posting a thread, would you please be so kind as to include more substantial comments as '?'.
I know you may be confused, but if you're just going to say that, you might as well say +1 - so that instead of being a spamming idiot, we just think you're spamming.
Oh, +1.
The thread will shortly be disregarded and/or locked.

Sorry, there was actual content to the posts
that currently are listed as '?'.

I just didn't know how to delete them properly.

[Aeson]

I have never taken a pack of cards and thrown them out of an airplane and then verified that they didn't re-order themselves. Does that mean it's statistically impossible that they will not re-order themselves?

[Diadem]

Stop spamming for heaven's sake Enigma_Nova. We've seen the dancing lock by now.

Why should this thread be locked? It's very interesting, even if it's offtopic.