Studying probability right now and seeing how the powerset mobius function immediately implies the inclusion-exclusion formula is pretty cool. Thanks for this!

The Möbius function actually appears quite often in statistics/probability theory. I wrote more about this in the paper (https://arxiv.org/pdf/2404.14423), but in short: if you invert moments with the powerset Möbius function then you get central moments, but if you invert moments with the partition Möbius function, then you get cumulants. In fact, you can vastly generalise this by changing the mereology from partitions to ordered partitions etc.