Changing our mind in the light of the observations

Putting aside box B$_{E}$, from which there is little to learn for the moment, we proceed with our `measurements' on box B$_?$. Imagine now that the first extraction gives White. There is little doubt that the observation has tochange somehow our confidence on the box composition and on the color that will result from the next extraction.

As far as the box composition is concerned, $B_0$ is ruled out, since “this box cannot give white balls,” or, as I suggest, “this cause cannot produce the observed effect.” In other words, hypothesis $B_0$ is `falsified,' i.e. the probability we assign to it drops instantly to zero. But what happens to the others? The answer of the large majority of people, with remarkable exceptions (typically senior scientists), is that the other compositions remain equally likely, with probability values then rising from 1/6 to 1/5.

The qualitative answer to the second question is basically correct, in the sense that it goes into the right direction: the extraction of White becomes more probable, “because $B_0$ has been ruled out.” But, unfortunately, the quantitative answer never comes out right, at least initially. In fact, at most, people say that the probability of White rises to 15/25, that is 3/5, or 60%, just from the arithmetic of the remaining balls after $B_0$ has been removed from the space of possibilities.

The answers “remaining compositions equally likely” and “3/5 probability of White” are both wrong, but they are at least consistent, the second being a logical consequence of the first, as can easily be shown. Therefore, we only need to understand what is wrong with the first answer, and this can be done at a qualitative level, just with a bit of hand waving.Imagine the hypothetical case of a long sequence of White, for example 20, 50 or even 100 times (I remind that extractions are followed by re-introduction). After many observations we start to be highly confident that we are dealing with box $B_5$, and therefore the probability of White in a subsequent extraction approaches unity. In other words, we would be highly surprised to extract a black ball, already after 20 White in a row, not to speak after 50 or 100, although we do not consider such an event absolutely impossible. It is simply highly improbable.

It is self-evident that, if after many observations we reach such a situation of practical certainty, then every extraction has to contribute a little bit. Or, differently stated, each observation has to provide a bit of evidence in favor of the compositions with larger proportions of white balls. And, therefore, even the very first observation has to break our symmetric state of uncertainty over the possible compositions. How? At this point of the discussion there is a kind of general enlightenment in the audience: the probability has to be proportional to the number of white balls of each hypothetical composition, because “boxes with a larger proportion of white balls tend to produce more easily White,” and therefore “White comes easier from $B_5$ than $B_4$, and so on.”