Sapienza trust model derivation showing equivalence with random answers

So far we have stated that the trust model in Sapienza ( https://ceur-ws.org/Vol-1664/w9.pdf) is the same as modeling the untrustworthy part of a source as random. What does this mean? If I trust my source 90%, then 10% of its reporting is untrustworthy, meaning it answers the question randomly for that 10%.

Let's take a look at some examples and then try to derive a general equation which we will then equate to the trust equation in Sapienza.

We will use the real/fake node case because it is simple to follow. We have 100 new nodes and a source that reports with 60% confidence that the nodes are real (or fake). For purposes of review we'll take the case of 100% trust first. For a single source we would now have a confidence of 60% that the node is real. We just believe the source and no calculation is required.

For two sources at 60% we can model the situation as follows:

100 N
        60%        60%
50 R ==> 30 Tr ==>  18 Tr  (ie, for 50 Real nodes, our first 60% confident source reports 30 to be real (Tr) and 20 to be fake (Tf). Our second 
source independently does the same.
                   12 Tf

        20 Tf ==>  12 Tr
                    8 Tf


50 F ==> 30 Tf ==>  18 Tf (same as above for 50 Fake nodes. It is important to note that if our source is 60% confident about real nodes it is also 
60% confident about fake nodes)
                   12 Tr

        20 Tr ==>  12 Tf
                    8 Tr

This is two 60% tests in a row. If two tests in a row say it's real, what is the probability of it being real? Ie how confident should we now be?

18+8 nodes tested real twice. Of those 18 are actually real ==> 18/(18+8) = 0.692. Now we are 69.2% sure the node is real. This is the same, btw, as the Bayes eqn in Sapienza: 0.6*0.6/(0.6*0.6+0.4*0.4) = 0.692.

Now let's build the trust factor into the eqn. Let's suppose I trust Source 1 90%. This means 10% of the time my source will report randomly that the node is real or fake no matter what their test says it is.

SOURCE 1 WITH TRUST:

100 Nodes With Trust
         S1 60% confidence 90% trust (90% of reported answers are same as test and 10% are random)
                                                   Totals
50 R ==> 30 Tr (27 rTr, 1.5 rTr, 1.5 rTf)   ==>    29.5 rTr                                                 
         20 Tf (18 rTf, 1 rTr, 1 rTf)              20.5 rTf  
                                              

50 F ==> 30 Tf (27 rTf, 1.5 rTf, 1.5 rTr)   ==>    29.5 rTf
         20 Tr (18 rTr, 1 rTr, 1 rTf)              20.5 rTr