Is probability objective?

Question

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?

Answer 1

I think you start defining a problem with a suitable probability space. Each probability space is constructed depending on a certain phenomenon connecting each sample with a certain probability in the range $[0,1]$.

Although the sample space can be the same for these two problems, namely $\{\text{white ball},\text{black ball}\}$ the probability measure defined from $P:\mathcal{F}\rightarrow[0,1]$ is different. Therefore, each problem is different.

I think we can not talk about objectivity as long as we are working on the same probability space. Because apperantly these two problems are distinct.

In such a scenario, everybody is correct if I must answer your question 'who is correct?'.

Answer 2

If Bob know which ball he took why do you think the probability is still interested?

Answer 3

I refer everyone to this discussion.

I disagree with "celtschk"'s comment (and his spelling of "Bayesian") that "both are right". If I ask what is the probability that there was life on Mars a billion years ago, and answer that the conditional probability, given all that I know, is $1/2$, I am behaving like a Bayesian. If say that the probability that a die gave me a "1" the first time is was thrown is $1/3$, given my knowledge that a "1" resulted in $1/3$ of all throws in a sequence of $6$ million trials, conjoined with my ignorance of any other relevant information, I am again behaving like a Bayesian. If I say that the probability of a "1" on the first trial, given my knowledge of the whole record of $6$ million trials, is $1$ (since the record indicates that that was the result on the first trial), I am again behaving like a Bayesian. A Bayesian treats probabilities as epistemic, not as frequencies. Frequencies can be relevant information on which probabilities are based, but from the Bayesian point of view, they are not themselves the same thing as probabilities.

Answer 4

I agree with Seymour but like to think of it a different way. You have one problem but Alice's answer represents a conditional probability and so was Bob's. But they condition on different events due to having different knowledge about how things changed when bob ate the ball. The probability can still be viewed as objective.

Whenever you talk about subjective vs objective probability it is likely to trigger a discussion of the Bayesian approach to probability ala De Finetti. But this question has nothing to do with that.