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...For example, flipping a coin ten times in a row and getting heads each time is very unlikely; one would only expect it to happen about 1 in 1,024 tries. Most of us would not sit around flipping coins just to see it happen, but if we had a million people flipping coins, we would see it happen many times. This phenomenon is publicized in the newspapers when lottery winners are announced. Winning a million-dollar jackpot is unlikely, but with millions of people purchasing tickets, eventually someone wins...

Using the 10 heads in a row analogy stinks. It has to assume that once you flip heads 10 times in a row, it will now always be heads on any flip. Randomness can no longer apply

Rather, try this one on for size.

Take 10 identical coins and mark them 1 to 10.
Place them in your pocket.
Now take one out...there is one chance in 10 that you will get the number 1.
Now put it back in your pocket.
Pull a coin. The chances that 2 will follow 1 are not 1 in 10 , but 1 in a hundred.
With each new coin taken out, the risk is multiplied by 10.
So that 10 following 9 is 1 in 10,000,000,000.

Now try that with all the things that are required to make life.

Randomness ? Not a chance.

It has to assume that once you flip heads 10 times in a row, it will now always be heads on any flip

Not so, but a "fake coin" would have this property, but a real coin can "randomly" create any pattern.

Take 10 identical coins and mark them 1 to 10. Place them in your pocket. Now take one out...there is one chance in 10 that you will get the number 1.

Agreed

Now put it back in your pocket. Pull a coin. The chances that 2 will follow 1 are not 1 in 10 , but 1 in a hundred

I don't understand here, are you trying for a two on a single draw? Or are you now trying for a 1 - 2 draw in sequence. If the latter I agree it will be 1/100 to have this occur. But if you are trying for a 2 (ignoring the first draw) then the odds are one in ten just like they were for drawing a one. The coins have no idea what you drew the time before.