I don't have a summary offhand that would be better than anything you can Google up, but this post by Sean Carroll explains the issue and details one possible answer: http://www.preposterousuniverse.com/blog/2014/07/24/why-prob...

I don’t have a real source, but the basic problem is somewhat straightforward. Suppose I have a particle in the state |0> + |1>. (I’m ignoring overall normalization.). After the measurement, the state is (|0>|I measured 0>) + (|1>|I measured 1>). This is a pure (deterministic) state.

It would be nice to say that there’s a 50% chance that I measured 0, but how exactly do you get that in a rigorous way from the state vector above?

To make everything complicated, the answer should not treat the experimenter part of the universe specially.

Your second state is only pure if no information has leaked into the environment. The chance of a human-sized object measuring a state without any stray photons or air molecules interacting is basically zero.

As for why 50%, why not the Born rule? Or are you asking how we derive the Born rule?

> Your second state is only pure if no information has leaked into the environment.

Not true in a many-worlds model. The “I measured” part is intended to account for the environment, at least initially.

> Or are you asking how we derive the Born rule?

More or less. In a many-worlds interpretation, there is no Born rule per se. I’m saying it’s not entirely trivial to recover the statistics that the Born rule would give.