[danbruc]

The bet does not really matter, the central question is whether they have a fair coin or if they are trying to me in some way. Even without the magic store, I would be very suspicious of anyone approaching random people with an offer like that.

So I would certainly consider it likely, that they are trying to trick me. But the probability I would assign to this, would still be rooted in some frequency, somewhere under the hood I would try to estimate the possible situations leading to such an offer and in which fraction of them I will be tricked.

If I am doing a good job with that, then repeatedly being in this situation should result in me getting tricked with the probability I cooked up. If I am bad at figuring out the possible states and their probabilities, then I they will not match.

[glial]

The key operation of Bayesian inference is integrating information. This can be from the same source (an additional coin flip, for example) or from different sources (coin flips, plus auxiliary knowledge about where the coin came from, or the person's motives).

Calculated frequency is a point-estimate of bias. A Bayesian estimate is a distribution of belief over possible values of the bias, integrating all available information.