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.