I think we are actually looking at the problem from different perspectives and that is what is confusing us. I am specifically talking about the situation where we are looking for one exact outcome thus order matters. You are looking at it from the larger picture. So you are right all different combinations are equal but if you are looking for a specific combination, what I am talking about, then it is less likely due to the other possibilities being erroneous. So in your scenario each flip is separate in mine they cannot be since we are only looking for one outcome.
avaserfi, we certainly might be looking at this differently. However, at the moment my perception (which has been wrong many times) is that you think that some specific combinations are more likely than others. That's not the case, and that doesn't agree with how many of us think. I know, because it took me a while to understand this back when I was learning this stuff.
You are completely correct in that one specific outcome is not as likely to occur as it is to not occur (provided that more than two outcomes are possible). Like in my three-coin-flip example, the probability of getting all heads is 12.5% whereas the probability of not getting all heads is 87.5%.
My two points are these, though:
1. The probability of getting all heads is exactly the same as getting any other specific combination (such as heads-tails-heads)...they are all 12.5%.
2. The probability of getting heads on any coin toss is always...always...50%, regardless of what has happened before or what you are looking to happen in the future. The probability of getting all heads over the span of 1000 coin tosses does not in any way affect the 50% probability of any given toss coming up heads.
Are we on the same page? Probably, but like you said, we're just saying things differently.