ADHD and autism are comorbid in one direction (people with autism are more likely than the general population to have ADHD, but people with ADHD are not more likely than the general population to have autism).
> people with autism are more likely than the general population to have ADHD, but people with ADHD are not more likely than the general population to have autism
I'm pretty sure that's not mathematically possible. If W people have both, X people have only autism, Y people have only ADHD, and Z people have neither, can you provide values for W, X, Y, and Z that would make that claim true?
Our assumption is that the ratio of people with autism and ADHD to people with just autism is greater than the ratio of people with ADHD to people in general. Using your variables, that is: W / (W + X) > (W + Y) / (W + X + Y + Z).
This simplifies out to the constraint that ZW > XY.
We want to show that the ratio of people with autism and ADHD to people with just ADHD is no greater than the ratio of people with just autism to people in general. This is: W / (W + Y) <= (W + X) / (W + X + Y + Z).
We can recognize that this is almost the same as our assumption, just exchanging the roles of X and Y. (I guess that's obvious from the problem statement.) That means our goal is to show that ZW <= YX, which is immediately just the negation of what we're assuming. That would be a contradiction.
So, indeed, this is mathematically impossible!
(EDIT: It's surprisingly hard to write these formulae down in English without getting confused.)
I think something that would be possible is that having ADHD if you have autism is more likely than having autism if you have ADHD. For instance, "being older than 20 if you're older than 30 is more likely than being older than 30 if you're older than 20" is just obviously true, because one entails the other.
We don't need mathematical or conjectural proofs here, it makes intuitive sense but if you find a soecific rebuttal to it we'll be glad to reference that as well
