WHERE IS PROBABILITY?

The most important outcome of the discussion related to the toy experiment is in my opinion that, although people do not immediately get the correct numbers, they find it quite natural that relevant changes of the available information have to modify somehow the probability of the box composition and of the color resulting in a future extraction, although the box remains the same, i.e. nothing changes inside it.Therefore the crucial, rhetorical question follows: Where is the probability? Certainly not in the box!

At this point, as a corollary, it follows that, if someone just enters the room and does not know the result of the extraction, he/she will reply to our initial questions exactly as we initially did. In other words, there is no doubt that the probability has to depend on the subject who evaluates it, or

“Since the knowledge may be different with different persons or with the same person at different times, they may anticipate the same event with more or less confidence, and thus different numerical probabilities may be attached to the same event.” (6)

If follows that probability is always conditional probability, in the sense that

“Thus whenever we speak loosely of `the probability of an event,' it is always to be understood: probability with regard to a certain given state of knowledge.” (6)

So, more precisely, $p=P(E)$ should always be understood as $p=P(E\,\vert\,I_S(t))$, where $I_S(t)$ stands for the information available to the subject $S$ who evaluates $p$ at time $t$. It is disappointing that many confuse `subjective' with `arbitrary', and they are usually the same who make use of arbitrary formulae not based on probability theory, that is the logic of uncertainty, but because they are supported by the Authority Principle, pretending they are `objective'.