Although there is some resemblance in that OP formula to that for parallel resistance formula, that is purely maths coincidental and the application it is of has nothing to do with parallel resistance (or at least, with my limited imagination, I cannot see how the two could meaningfully relate).
In the particular application I used this OP result in my current research, in the OP, a and b and c and d are all possible unknown probability values for k of a categorical distribution having k number of categories except, and unlike what is implied by my OP, there isn't just merely 4 possible categories to this distribution but an infinite number of them that become progressively smaller so that the sum up to 1 (else this distribution OF a categorical distribution will not normalize).
But what you would know is what the relative proportions
of y/a and y/b etc must be where y is the probability of the distribution having k=c categories where c is the total number of sampled categories so far (there may be more categories not yet sampled so actual unknown k may be greater than c).
There are other parameters I didn't mention in the OP but, with huge difficulty and this took my several days, the equation for y I eventually derived was
y = k! (k – 1)! / ( (k + m – 1)! ∑[n=0, ∞] (k + n – 1)! (k + n)! / ( n! (k + m + n – 1)! ) )
where k=c and where
m = the number of samplings of a/any category (whether of the same category or a different category).
k = number of categories of distribution
c = total number of sampled categories so far
y = probability of number of categories of distribution being the same as that of total c number of sampled categories so far
I just bet all the above info would do far more to inadvertently confuse you all rather than inform you!