This isn't a "statistics" problem exactly, but rather a data science problem, but... Personality tests are usually framed as psychological instruments, but the core problem is statistical. They assume responses are noisy observations of a stable latent variable, when in practice they’re samples from a context-dependent decision process. Respondents condition on incentives, infer what’s being rewarded, and optimize accordingly, so the data-generating process shifts with each setting. From a statistical perspective this breaks identifiability: multiple latent personalities can produce the same observed responses, and the same personality can produce different responses under different incentive structures. Because those incentives are unobserved or only weakly observed, the model is fundamentally misspecified. More samples don’t converge on truth; they converge on a biased estimate of a strategic equilibrium rather than the latent trait itself.

On paper, I'm a real collaborator, because that is what I think the test should reveal about me. In reality, I just don't wanna deal with collaboration and I'd rather work alone. Can this be measured? Probably not, but it does present an odd paradox for data scientists to solve.

Here's one I did at A-level: https://en.wikipedia.org/wiki/Bertrand_paradox_(probability)

It sounds like you're looking for problems in probability theory (rather than statistics). I don't have anything specific for you but you might have better luck searching for problems, puzzles, and examples in probability. For instance:

https://math.stackexchange.com/questions/2140493/counterintu...

Look into a concept called "The curse of dimensionality". https://en.wikipedia.org/wiki/Curse_of_dimensionality

Simpsons paradox and anscombe quartet are pretty good starters