As we know, probability is a measure of events. However, is it an objectively attribute of events, or just an illusion in ones' mind?
For example, suppose that there is an empty black box with an autoclosing cover in its top-side. And 2 players: Alice and Bob. Then
1)Alice put a white ball and a black ball into the box. Bob saw the process.
2)Alice closed her eyes.
3)Bob took out one ball and ate it up. He knew which ball he took whereas Alice did not know(but she knew Bob had taken one ball).
4)Alice opened her eyes.
5)They took a paper respectively, and wrote down the probability of 'the box has a white ball'.
Now we are sure that numbers they wrote are different, Alice must wrote a number equals to 0.5 whereas Bob's was either 0 or 1.
So my question is 'who is correct?' And furthermore if both are right then does one event can has more than one probability at the same time? If only one is right, why the other is wrong?