[jbay808]

A 1 in 1024 chance is not an absurd premise! Something that has a 1 in 1024 chance of ocurring to a person happens to 7 million people. If you get that evidence, you need to be capable of interpreting it correctly.

The OP's example would also worked well enough with even just 5 heads; you'd already pass p < 0.05, despite the actual probability of having picked the biased coin still being minuscule.