one run is more likely than the other, thats a mathematical fact.
No, that's not true.
Coin no. 10 has no knowledge of what the previous 9 coils fell, so coin 10 has a 50/50 chance of coming up heads, just as coins 1, 2, 3 etc do.
HTHTHTHTHT is exactly as likely as HHHHHHHHHH or HHHHHTTTTT
But we are talking about a sequence or run of events. Yes each coin is 50/50.....but what is the probability, thats how we work it out.
Flip 10 coins...HHHHHHHHHH is 1/1024
Flip 10 coins...HTHTHTHTHT is 252/1024
Flip 10 coins...HHHHHTTTTT is 252/1024
Flip 1 coin...H is 1/2
So to flip a coin 10 times, with a result of all heads, as a sequence not a single event, is very unlikely.
Each of those three sequences are specific sequences so all thre are 1/1024.
The chances of getting 5 heads in a run of 10, there position within that sequences being unimportant, THAT is more likely than any of the sequences mentioned above, but once you state what the position is, then the odds are 1/1024.
That is true yes, I was just choosing random HT combinations, any combination containing 5 heads and 5 tails will be 252/1024, and so on.