[monktastic1]

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?

[amluto]

> 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.