One in thirteen - Bayesian reasoning illustrated with a toy model

Imagine there are two types of boxes, , that only contain
white balls (), and , that contain one
white balls and twelve black
(incidentally, just to be precise,
although the detail is absolutely irrelevant,
we have to infer from Columbo's words,
*``You didn't touch any
of these twelve cameras. You picked up that one''*,
the cameras were thirteen).

You take at random a box and extract a ball.
The resulting color is
*white*.
You might be interested to evaluate
the probability that the box is of type ,
in the sense of stating in a quantitative way
how much you believe
this hypothesis.
In formal terms we are interested in
,
knowing that
and
, a problem that can be sketched as

(1) |

[Here `' stands for `given', or `conditioned by'; is the general (`background') status of information under which this probability is assessed; `' or `' after `' indicates that both conditions are relevant for the evaluation of the probability.]

A typical mistake at this point is to confuse with , or, more often, with , as largely discussed in Ref. [1]. Hence we need to learn how to turn properly into using the rules of probability theory.