A simple example of Bayesian vs. frequentist probability

Alice picks up a penny, puts it behind her back where Bob can’t see. She asks, “What’s the probability that the penny is in my left hand?”

For Alice, the answer is simple: the probability is 100% if the penny is in her left hand and 0% if it’s in her right hand. For Bob, the answer is tricky. The intuitive answer is 50%, as he has no knowledge about what hand the penny could be in.

I think this is an easy example of thinking about Bayesian versus frequentist probability. For Alice, who has perfect information, the two are the same: the Bayesian and frequentist probabilities are both 100% or 0%, depending on which hand she put the penny in. For Bob, the subjective, Bayesian probability is 50%, but the frequentist probability is the same as it is for Alice, either 0% or 100%.

Bayesian probability is subjective, so it can differ between Alice and Bob, but frequentist probability is objective, so it cannot be different.