Probability - How does it work?!

I find it hard to see what you are getting at. Maybe you can clarify.

By saying this:

"The problem (as usually stated) is what are the chances of flipping a head 100 times in a row. Rather than in an arrangement of 100 coins what percentage of possible arrangements feature XXXXX (XXXX being whatever you are measuring)"

I am saying that the problem as stated is asking for the probability of a set sequence (in this case a flip resulting in heads 100 times in a row) rather than the probability of the appearance of one of many sequences that share the ratios (H v T) of the original but not the order.

As Teome suggested, if you think of it as sequences, in a set of 100 restricted coin tosses the chances of ending up with 1 tail at some random point along the line and 99 heads is 100 times more likely (given that it has 100 different sequential possibilities than the single sequence possibility of all 100 heads).

It is a 100 times more likely because we are comparing 1 sequence to 100 sequences.

If this is correct (and unless I've misread, I think we're all agreed it is) then I dont see the relevance of your continuing argument about all sequences being *individually *equal in possibility.

Because they are. And because a sequence of 100 heads only has one order, there are no variations of 100 heads in a row.

Contrary to what you are claiming, I would say that Teome has phrased the argument in the manner in which we face it in everyday life: When we say that 100 heads or 100 tails is near impossible, we don't mean it is less likely as an individual sequence than 99 heads with specifically the second toss being a tail, or 99 heads with the 55th toss being a tail, etc. What we normally mean is that the chance of picking out that one random sequence is astronomically smaller than simply betting on ANY of the massive massive number of other undisclosed random sequences.

You have this part wrong.

The chances of that sequence (100 Hs in a row) occurring from a random selection is identical to ANY of the massive massive number of other sequences.

All of them are 1 in 1,267,650,600,228,229,000,000,000,000,000.

A hundred heads = 1 in 1,267,650,600,228,229,000,000,000,000,000
95 heads with 5 tails in the last 5 places = 1 in 1,267,650,600,228,229,000,000,000,000,000
46 heads and 54 tails bunched at either end = 1 in 1,267,650,600,228,229,000,000,000,000,000
HTHTHTHTHTHTHTHT . . . = 1 in 1,267,650,600,228,229,000,000,000,000,000
TTTHHHTTTHHHTTTHHH . . . = 1 in 1,267,650,600,228,229,000,000,000,000,000
One head at the start, one at the end and 98 tails in between = 1 in 1,267,650,600,228,229,000,000,000,000,000

The likelihood of ANY of the massive number of sequences (1.267 nonillion) occurring is identical.