Why Does Spin Add? Quantum Mechanics Explained

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Why is it that spin adds? I mean sure, the classical analogy tells us that angular momentum is additive, but that I feel is a cop out. What inside of the formalism for quantum mechanics tells us that we should be adding spin of two particles?
 
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It doesn't. You can add spin 1 and spin 1 and get spin 1. Or 3 and 4 and get 2. Or...
 

Thread 'Is a table levitating possible, even at a minuscule probability?'

I am not sure if this falls under classical physics or quantum physics or somewhere else (so feel free to put it in the right section), but is there any micro state of the universe one can think of which if evolved under the current laws of nature, inevitably results in outcomes such as a table levitating? That example is just a random one I decided to choose but I'm really asking about any event that would seem like a "miracle" to the ordinary person (i.e. any event that doesn't seem to...
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