Ideas, beliefs and probability

In other terms, finally calling things with their name, we are talking about degree of belief, and references to the deep and thorough analysis of David Hume are deserved. The reason we can communicate with each other our degrees of belief (“I believe this more than that”) is that our mind understands what we are talking about, although

“This operation of the mind, which forms the belief of any matter of fact, seems hitherto to have been one of the greatest mysteries of philosophy
...
When I would explain {it}, I scarce find any word that fully answers the case, but am obliged to have recourse to every one's feeling, in order to give him a perfect notion of this operation of the mind.”  (8).

In fact, since “nothing is more free than the imagination of man” (9), we can conceive all sorts of ideas, just combining others. But we do not consider them all believable, or equally believable: “An idea assented to feels different from a fictitious idea, that the fancy alone presents to us: And this different feeling I endeavour to explain by calling it a superior force, or vivacity, or solidity, or firmness, or steadiness.” (8) (italics original.)

An easy evaluation is when we have a set of equiprobable cases, a proportion of which leads to the event of interest (neglect for a moment the first sentence of the quote):

[“Though there be no such thing as Chance in the world; our ignorance of the real cause of any event has the same influence on the understanding, and begets a like species of belief or opinion.”]
“There is certainly a probability, which arises from a superiority of chances on any side; and according as this superiority encreases, and surpasses the opposite chances, the probability receives a proportionable encrease, and begets still a higher degree of belief or assent to that side, in which we discover the superiority. If a dye were marked with one figure or number of spots on four sides, and with another figure or number of spots on the two remaining sides, it would be more probable, that the former would turn up than the latter.” (9).

This is the reasoning we use to assert that the probability of White from box $B_i$ is proportional to $i$, viz. $P(\mbox{W}\,\vert\,B_i,I)=\pi_i$. Instead, the precise reasoning which allows us to evaluate the probability of White from $B_?$ in the light of the previous extraction was not discussed by Hume (for that we have to wait until Bayes (10), and Laplace for a thorough analysis (11)), but the concept of probability still holds. For example, after four consecutive white balls the probability of White in a fifth extraction becomes about 90%. That is, assuming the calculation has been done correctly, we are essentially so confident to extract White from $B_?$ as we would from a box containing 9 white balls and 1 black.