I was wondering if its possible to solve this question in terms of $x,y,z$ (parameters, instead of actual numbers)/
We have a bag of red and blue balls. Suppose red count is $x$ and blue count is $y$. Someone else, let's say David, takes one ball, observes the color and returns it into the along with $z$ more balls of the same color. I don't have any information about what David saw or did. I pick a ball at random from the new bag, and observe red. Question what's the chance that David saw a red ball as well, in terms of $x, y, z$?
Is it possible to derive a general formula parametrically in terms of $x, y, z$?
Yes, it's possible.
For instance, you know the probability that "I picked red if David did" is: $$\mathsf P(I\mid D) = \dfrac{x+z}{x+y+z}$$
Because if David had picked red there will be z more red balls in the jar when "I" pick. And so forth for "I picked red if David did not", "David picked red" and "David picked blue".
Then you can use such probabilities to evaluate the probability that "David had picked red given that I did" by using Bayes' rule (and the Law of Total Probability).
$$\mathsf P(D\mid I) = \boxed ?$$
Have a go at it.