I still want to hear your reasons for doubting KS as an appropriate method, but now that I've looked at your graph more, I say that KS does not apply. Your data are discrete, and KS does not apply to data drawn to discrete distributions. However, you could use a chi-squared test! I wrote about this yesterday. Instead of checking if frequencies match the ...
Results will depend on the numbers of non-v events in each group.
If data are as below, with counts of v in a and non-v in b, so that the total number of events in each group is 100, then the null hypothesis that probabilities of v are homogeneous among groups is overwhelming rejected
with P-value nearly 0. The test is a chi-squared test of homogeneity. (...
For the comparison of experimental results with a theoretical probability
distribution to make sense, both need to refer to the same thing measured
the same way. Suppose you're talking about getting at least three Aces
when fairly dealt five cards from a well-shuffled standard 52-card deck.
Theoretical: Let $X$ be the number of Aces obtained:
$$P(X \ge 3) = ...