Statistically, the results are the same.
100% x $100 = (50% x $0) + (50% x $200)

What isn't the same is the spread of the results.
In the first case, I'd ALWAYS lose $100.
In the second, I'd lose $100 ON AVERAGE.

So, I'd think gamblers would tend to go with choice #2, since they hope for the best.
More conservative people would go with choice #1, since they'd fear the worse.

An opportunist would bop you in the jaw while you are watching the coin go up, and take all your money.