I guess I wasn't clear. I was talking of the prior probability of that sequence having occurred. It is about 1 in 10^300.
It's not like you flipped your coin millions of times until you saw that particular (and no other) sequence to post it to me. You simply flipped your coin 1,000 times and sent the results to me. The odds of your result appearing, since you had no pre-concieved notion of what it would be, are 1 in 1.
The odds of that sequence re-appearing, may very well approach 1 in 10^300. What you are trying to do is to claim that BOTH the sequence appearing and re-appearing are slim. That's simply not the case. Flip your coin and you are going to get an output that you can send me. The odds of that output appearing are 1 in 1 until you pre-specify your output condition, wherein the odds then move towards your 1 in 10^300 figure (presuming that your math is valid, which for this particular concept hardly matters).
Put another way, the odds are 1 in 1 that you will have an output to send to me if you flip a coin a thousand times. The odds that a pre-determined particular sequence will appear, however, will approach your 1 in 10^300 figure.