How can randomness possibly exist?

[Borodog]

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Ok, I'm slow. Apparently the atomic clock does not measure time based on the decay of atoms, but instead based on the wavelike movement of electrons inside the atom. Can someone just confirm this for me to make sure I'm understanding it correctly?

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Not quite. When an electron in an atom drops to a lower energy level, radiation is emitted. That radiation has a specific frequency. Combined with the (constant) speed of light, this provides the most accurate known measures of time.