One reason why a lot of things are comorbid: generally, few mental things have "a single gene" but are manifestations of many gene differences and nature.

One reason why a lot of things are comorbid is just that more people are a little something than a lot something, and people who get diagnosed for one thing are more likely to have resources and reasons to be diagnosed for another something.

After rereading your msg, I'm concerned you're looking for some sort of thing like "If I'm sufficiently good at X I can't be Y"

I don't think these things work this way. Many folks on the spectrum share some social skills weaknesses, but social skills challenges are not really what makes them on the spectrum.

In all, all models wrong - some models useful. An attitude of viewing these things from the lens of asking if they help you understand yourself and navigate the world is probably best. I was really worried with labels before I kept repeating to myself 'all models wrong, some models useful'

Depression, ADHD, autism, all models. All 'wrong'.Still, useful sometimes. They aren't meant to be the be all end all though, and they only sometimes (maybe often) accurately represent reality, which is a bit of a silly statement, since there isn't one reality to it all.

There we go, I got more confusing sorry :)

My understanding of the term 'spectrum' in this context is less "a scale from 0-100%" and more "a collection of traits which, if you have enough of them, would suggest you fit into this larger category".

In terms of being being able to navigate social situations, look up 'masking'. Many people on the spectrum learn to behave (sometimes very effectively) in 'normal' ways in social situations but it's a set of learned scripts and behaviours rather than being instinctive.

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.

People with autism are highly likely to have comorbid ADHD.

People with ADHD are more likely than normal to have autistic traits, although nowhere near as commonly as the first statement.

General chance of being in either group: 50% (10 people with autism, 10 with ADHD, 20 total).

Chance of having autism if you have ADHD: 50% (9 with autism, 9 without, 10 total).

Chance of having ADHD if you have autism: 90% (9 with ADHD, 9 without, 10 total).