First of all, how often you're correct doesn't matter; say the tails option wasn't "you'll be woken up a million times", but instead "you'll be asked the same s

First of all, how often you're correct doesn't matter; say the tails option wasn't "you'll be woken up a million times", but instead "you'll be asked the same stupid question a million times" (and have to answer as if you'd forgotten the previous answer). Of course you'll be wrong more often in the tails situation. Second, the experiment isn't run 10 times. It's run once. A single coin flip. With a multitude of flips, you've got no idea where you are in the sequence so it pushes your averages toward the 1:1000000. But a single coin flip is a single coin flip, 50/50.