Yet another newbie question. What's the rational way to behave in a prediction market where you suspect that other participants might be more informed than you?

Here's a toy model to explain my question. Let's say Alice has flipped a fair coin and will reveal the outcome tomorrow. You participate in a prediction market over the outcome of the coin. The only participant besides you is Bob. Also you know that Alice has flipped another fair coin to decide whether to tell Bob the outcome of the first coin in advance. What trades should you offer to Bob, and what trades should you accept from Bob?

Bonus points for solving a similar toy model where Bob has a 50% chance of influencing the outcome of the coin instead. Both questions seem relevant to real-world prediction markets, and also other betting situations.