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I assume there are worlds in the MWI where extremely improbable things (improbable according to our world) have happened repeatedly. Broken eggs reform themselves repeatedly etc... Will scientists in these worlds come up with the same laws of physics as this world? specifically the same laws of quantum mechanics? Assuming they follow the same scientific process we do...

How do we know we aren't in an "extreme" world? ie: How do we know our history of observations somehow matches the distribution of observations in different worlds...

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    $\begingroup$ maybe it's best to let to (bad) SF the idea of infinite parallel worlds each changing by some detail. Or obviously it leads to various paradox like this one and others. $\endgroup$ – Fabrice NEYRET Feb 16 '16 at 23:12
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    $\begingroup$ MWI is a good lesson for what can happen when you don't just take one bad intellectual turn but several in a row... just like one of those unlikely worlds. :-) $\endgroup$ – CuriousOne Feb 16 '16 at 23:52
  • $\begingroup$ I don't understand why MWI fascinate so much people ! perhaps, this means that they feel something I don't ... $\endgroup$ – user46925 Feb 17 '16 at 10:33
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The way the MWI treats probability is roughly as follows. You take the non-probabilistic part of quantum theory. You then assign decision theoretic payoff to each possible measurement outcome and assign a value to a state that respects non-probabilistic axioms of decision theory. One example of such an axiom would be that if two states $|\psi_1\rangle,|\psi_2\rangle$ are the same except that every outcome has an increased payoff in $|\psi_2\rangle$ then the payoff for $|\psi_2\rangle$ is higher than that of $|\psi_1\rangle$. David Deutsch has explained that you can derive the standard probability rule given those assumptions. The MWI can be tested using those probabilistic predictions. If a particular event would make your theory problematic then an experimental observation of that event is a criticism of your theory.

Now you ask about branches of the multiverse in which extreme events happen. Note that if this is a problem, then it is also a problem for theories in which there is a single universe and stuff happens randomly since unlikely stuff can happen in such theories. The main difference between the two cases with respect to probability is that the MWI explains the probability rule and the circumstances under which it is applicable, and the single universe theory postulates it arbitrarily. So any refutation of the MWI on that basis would refute all probabilistic theories. But in both cases, this amounts to saying that a measurement can go wrong, which can happen for other reasons. If you make such an observation, then you should check out a load of other sources of error before you consider the idea that you're in a very improbable branch of the multiverse.

Another part of your question is about whether we might see the wrong laws of physics, e.g. - the second law isn't really true. The problem is that this treats the second law as being an arbitrary postulate that could just as easily be completely different. But that is not the case, the second law is tied deeply to other parts of physics like the theory of information. See, for example, "Maxwell's Demon 2 Entropy, Classical and Quantum Information, Computing" edited by Leff and Rex. Replacing the second law requires replacing a whole load of other fundamental physics.

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How do we know our history of observations somehow matches the distribution of observations in different worlds...

We don't. All we know is that our history of observations generally matches the kind of statistical distribution we might expect from theory. [Not counting Hitler, of course!]

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