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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?

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If you are a Bayesianist, both are right. If you are a frequentist, it depends on whether, when repeating the experiment many times, Bob would choose the same ball each time (then Bob is right), would choose each ball equally often (then Alice is right), or would only show a certain bias for one color (then neither is right). – celtschk Sep 23 '12 at 13:51
@celtschk Er...I did not quite understand the last case(neither is right), would you please explain it? Besides, do you mean that the meaning of probability is different of various mathematician with various belief? – Popopo Sep 23 '12 at 14:00
Well, the last case would be for example if Bob eats the white ball twice as often as the black ball, the probability of the box containing the white ball is $1/3$, which is neither $0.5$ nor $0$ nor $1$. And yes, people disagree about the correct interpretation of probabilities. – celtschk Sep 23 '12 at 14:11
Probability is conditional: it depends on what information is available, and the information available to Alice and Bob are clearly different. Note that quantitative measures can be both relative and objective; for instance velocity depends on your frame of reference, but its actual value according to any given frame does not depend on a subject's feelings or intuition. Probability can mean both a subjective "degree of belief" or an objective description of a physical system as something whose behavior is empirically tied to what mathematical models predict. – anon Sep 23 '12 at 16:18
I disagree with "Celtschk"'s first statement. See my posted reply to this question. – Michael Hardy Sep 23 '12 at 16:40
up vote 2 down vote accepted

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?'.

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Do you mean that Alice and Bob met distinct problems, so they had different probability space? – Popopo Sep 23 '12 at 14:45
It is like looking at a mountain from different sides. You see one thing and I see another. Therefore I have a completely different problem than yours. Although we are all related to the same mountain.Therefore, we all do some reasonable things and we give some reasonable decisions related to our problems. Bob and Alice have the same place of problem but with different observations yielding a different phenomena. – Seyhmus Güngören Sep 23 '12 at 14:52
Hope I did not misinterpret what you mean. So the probability is depend on the phenomena and besides, for the all-seeing one, $P$ is always 2-valued? – Popopo Sep 23 '12 at 15:15
The phenomenon depends on the probability distribution with respect to your sample space. As long as they are different, this means you have two random variables with different distribution functions. They map to a different problem. $P$ depends on $\Omega$, your sample space. If $\Omega$ is discrete, then $P$ is a function from any collection of subsets of $\Omega$ such that they build a sigma algebra $\mathcal{F}$ on $\Omega$. Your distribution function $F(y)$ has two values on the $y$ axis. $F(y)$ can, however, be different depending on your phenomenon, although $y$ axis remains the same. – Seyhmus Güngören Sep 23 '12 at 15:24
So the probability distribution is depend on how complete the information cognitive subjects have? – Popopo Sep 23 '12 at 15:40

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.

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

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I think the problem can be manipulated such that Bobs observations have another distribution and still the idea of the question is there. – Seyhmus Güngören Sep 23 '12 at 14:55
It is because the two events are different in "math" – RHS Sep 23 '12 at 15:00
Do you mean that the major difference is not just in terms of $P$ , but also with $\mathcal F$? – Popopo Sep 23 '12 at 15:23 seems your observation also make a sense. – Popopo Sep 23 '12 at 15:43
Their sample spaces are different in the sense of conditional probability. The sample space contains all possible outcomes. And bob always have 1 possible outcome but not alice. – RHS Sep 23 '12 at 15:50

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.

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Okay, thank you. Did you mean that the probability is not depend on the epistemic situation but conditions do? – Popopo Sep 25 '12 at 15:18

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