I've read many times about an experiment: take a 2-particle system and measure that it has a spin of 0. This tells us that the particles have opposite spin. Now, take the particles far away from each other and measure one. If you measure spin up, for example, you now know the other particle has spin down.

Why would anybody be surprised by this?

Let's imagine a similar classical experiment that uses literal spin. Take a system of two gyroscopes, each spinning at the same fixed speed, each in it's own sealed box. They can be powered or whatever so they stay spinning for the duration. Stack the two boxes ("entangling" their spin), and measure 0 spin to confirm that the gyroscopes are in fact spinning in opposite directions (some helicopters use this principle for stabilization on the y-axis instead of a tail propeller). Now, send the boxes far apart and measure one of them using the right hand rule. If it turns out to have spin up, the other will intuitively turn out to have spin down. But nobody will be surprised by this because we knew from the beginning that the pair was spinning in opposite directions; we just didn't know which was which before measuring one of them.

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**Update: 2020Apr24**

Thanks for all the comments, explanations, and links! I'm not ignoring them, just trying to find time between work and family responsibilities to digest and understand them before I ask my follow-up questions. I appreciate your patience!

And see also Sidney Coleman's "Quantum Mechanics In Your Face" lecture (youtube, transcript) which walks through a cousin of Bell's theorem that's I think conceptually simpler—for example, it's a deterministic result, as opposed to a statistical correlation.