Yes, I agree that in principle every person should have a probability for each event. But what does the market tell us about what our own belief should be? If y

Yes, I agree that in principle every person should have a probability for each event. But what does the market tell us about what our own belief should be? If you don't have any nonpublic information and you trust the market to be efficient then your assumption should be that you yourself cannot make an expected profit in the market. So the useful information from the market can be given in terms of the bounds within which you can't bet profitably. You yourself should update your probabilities to be within those bounds.