Would you like to tell us precisely what your mathematics has proved. In simple terms please as I get confused. I once read a whole book on maths absolutely convinced it was The Operator j what dunnit only to find out on the last page it were Curly Dee. So you see my dilemma.
If you have two parallel but separate decisions to make, the probabilities of each test need to be multiplied together, reducing the accuracy of the two parallel tests.
If you have a series of tests, the second test improves the accuracy of the two tests.
So, if you have two blood dogs and bother alert in sequence, the probability of them being wrong is reduced from 4 in 5 to 4 in a hundred.
Consider the matrix for this case
Dog one on average is wrong once in five alerts as is dog two. We need to examine all possible permutations of correct and incorrect alerts.
Each dog alerts R R R R W
Showing this as a matrix we have
DOG TWO
DOG ONE.
W R R R R
W R R R R
R W R R R
R R W R R
R R R W R
R R R R W
There is only one place in the matrix where both are wrong so the probability is 96% correct increased from 80%
Now consider two different tests- see the previous descriptions for either Eddie and Keela, or the choice of spheres either red or blue and hollow or solid
The matrix now is
R R R R W
R R R R W
R R R R W
R R R R W
W W W W W
giving an accuracy of 16 out of 25 or 64%