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Repeated experiments have shown that the universe violates Bell's inequality and therefore does exhibit randomness.
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According to Trantor's post above it seems some physicists are still working on a hidden variable theory. I wonder why they weren't convinced!?
Based on all the responses so far I've clicked on some of your links and done some digging myself and I found a couple of interesting things:
1) A poll of 72 leading physicists conducted by the American researcher David Raub in 1995 and published in the French periodical Sciences et Avenir in January 1998 recorded the following results:
Yes, I think MWI is true 58%
No, I don't accept MWI 18%
Maybe it's true but I'm not yet convinced 13%
I have no opinion one way or the other 11%
link
2) Q13 Is many-worlds a deterministic theory?
Yes, many-worlds is a deterministic theory, since the wavefunction obeys a deterministic wave equation at all times. All possible outcomes of a measurement or interaction (See "What is a measurement?") are embedded within the universal wavefunction although each observer, split by each observation, is only aware of single outcomes due to the linearity of the wave equation. The world appears indeterministic, with the usual probabilistic collapse of the wavefunction, but at the objective level, which includes all outcomes, determinism is restored.
Some people are under the impression that the only motivation for many- worlds is a desire to return to a deterministic theory of physics. This is not true. As Everett pointed out, the objection with the standard Copenhagen interpretation is not the indeterminism per se, but that indeterminism occurs only with the intervention of an observer, when the wavefunction collapses. (See "What is the Copenhagen interpretation?")
link
So the many-worlds interpretation, which is in contradiction with the Copenhagen interpretation, looks like it's widely believed among leading physicists. If it's the correct one, then QM is a lot of things, but it's not evidence of inherent randomness in the universe.