Given that this is hacker news, I'm surprised that no one bothered to write a simulation to sanity-check their answer. Here's a quick and dirty one: http://jsfiddle.net/qc9qk/2/

The mistake that most people seem to be making is using P(biased) and P(fair) instead of P(biased|10 heads) and P(fair|10 heads).

Spoiler: The answer is ~0.75.