Basically, before you decide to assayed the urn (by keeping in mind the material of a money pulled from it), the likelihood it was of type 1 was about 66 percent
Figure 4c shows each of these same avenues more separated into two components, symbolizing the general percentage of coins which can be copper and silver in every one of two kinds of urns. Another role is of unit room (= 2/3 A— 7/10), showing the portion of coins being in both urn 1 and sterling silver. Another part is of device neighborhood 8/30 (= 1/3 A— 8/10), revealing the percentage of coins that are throughout urn 2 and copper. And also the last part was of unit location 2/30 (= 1/3 A— 2/10), showing the portion of coins that are throughout urn 2 and gold. As is viewed, P(U1&C) is available by multiplying P(U1) by Pm(C), thereby by multiplying the a priori chance that an urn are of sort 1 by the likelihood that a coin in an urn of kind 1 was copper (as per our original formulation of the complications). That is, P(U1&C)=P(U1) A— Pm(C), etc your some other combos.
Finally, provided this type of a priori possibilities and such likelihoods, that which you being asked to calculate is an a posteriori likelihood: the possibility that the urn is of kind 1 (or type 2) after you get a coin of a certain material (which alone constitutes a particular version of proof). This might be created as PC(U1), and so forth for other combinations. Figure 4d shows a geometric reply to this matter: Pc(U1) is equivalent to 6/14, or even the area P(U1&C) separated by sum of areas P(U1&C) and P(U2&C), that is equivalent to every methods for acquiring a copper coin from an urn of type 1 (6/30) separated by most of the means of getting a copper money whatever the brand of urn its drawn from (6/30+8/30). And once you assayed the urn, the possibility involved 43%. Or, phrased another way, ahead of the assay, your believe it had been very likely to end up being an urn of sort 1; and after the assay, you think its more likely to be an urn of sort 2.
Figure 5 is an additional method of showing the info for sale in Figure 4, foregrounding the algebra of the difficulties as opposed to the geometry, and thus 
iliar for most customers (though maybe reduced user-friendly). Figure 5:
As might seen, the key picture, most likely is claimed and completed, expresses the a posteriori probabilities with regards to the product with the likelihoods and the a priori possibilities:
One component are of device area 6/30 (= 2/3 A— 3/10), showing the portion of coins which are both in urn 1 and copper (thereby the intersection of coins in urn 1 and all copper coins)
Such a manner of creating the issue (usually described as Bayes‘ Rule), but processed or unimportant it could initial show up, actually is very general and strong. In particular, to come back to the issues of this preceding point, replace kinds of urns with sort; replace coins with indicator; and replace particular urns (which might be of a single sort or some other) with people. This way, we could possibly think of Bayes‘ Rule as a heuristic that a representative might adopt for attributing types to specific via their own indices, and so a method for changing unique ontological presumptions regarding the kindedness with the individual under consideration. In this way, the center picture, within its complete generality, might be shown below: