# One in thirteen - Bayesian reasoning illustrated with a toy model

Let us leave aside Columbo's cameras for a while and begin with a different, simpler, stereotyped situation easier to analyze.

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.

Subsections
Giulio D'Agostini 2010-09-30