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Square rooting will get you in the proper ballpark. I imagine that's why UUIDs are 128-bit values.

Correct on both counts.

Back of the envelope math for birthday paradox is root n. So in this case with ~50% probability you'll get a collision after 2^32 IDs (or after 4 seconds of 1 billion writes a second)

Ah, good point. I'm aware of the birthday paradox but I took (2^n)/2 = 2^(n-1) instead of 2^(n/2) as the birthday bound. Nice correction :)