As you're walking along the street in front of Honest Jake's Biased
Coin Factory, you find a quarter on the sidewalk. You suspect it is
biased, so you flip it 41 times and observe that it comes up heads 29
times. A bystander notices your interest in probability and offers $10
even money that your next two flips are not both heads. Do
you take the bet?

Clarifications and assumptions:

- Assume the coin is from Honest Jake's factory.
- Honest Jake guarantees that every flip of their coins is independent.
- Honest Jake makes coins with any bias and has no preference for any
particular bias. His motto is ‘My coins are biased, I'm not.’
- The bystander pays you $10 if the next two flips are heads, otherwise
you pay the bystander $10.
- You will be willing to take the bet if you expect to gain money and
unwilling if you expect to lose money.

I found this problem on the web (I'll credit it later). The problem was intended to show the difference between Bayesian and traditional reasoning, but the original poster made a slight error and got the wrong result. This problem has a very subtle point to it. I'll post the answer in a few hours.