Let's say somebody makes a prediction that something will happen, and it doesn't. You could evaluate the prediction differently if they gave it a 55% chance of happening vs a 95% chance.
I guess you could attach the probability as weight to the event outcome, instead of taking a uniform mean. Not that without probabilities it’s “impossible”, it’s just a different metric
How exactly do you propose to do that given the binary knowledge of the event occuring or not...