Some have argued that there is a way to get the Born Rule in MWI, but the mainstream opinion is that those arguments are circular. For example, see this recent paper:
How Do the Probabilities Arise in Quantum Measurement? Mani L. Bhaumik ...The author goes on to argue that he has solved these problems, and found a solution that has eluded physicists since 1926.
So far, only some ad hoc propositions such as Born’s rule  have allowed the physicists to predict experimen- tal results with uncanny accuracy of better than a part in trillion . But the basic cause of this essential rule has remained shrouded in a veil of mystery. One of the prominent investigators in this field, Wojciech Zurek has attempted to provide a derivation of the Born rule per- haps to make his program comprehensive . But it has faced a stiff resistance from some foremost investigators including one of the giants of physics of our time, Nobel laureate Steven Weinberg.
In his classic textbook, Lectures on Quantum Mechan- ics, Weinberg states [8, p. 92], “There seems to be a wide spread impression that decoherence solves all obstacles to the class of interpretations of quantum mechanics, which take seriously the dynamical assumptions of quantum mechanics as applied to everything, including measure- ment.” Weinberg goes on to characterize his objection by asserting that the problem with derivation of the Born’s rule by Zurek “is clearly circular, because it relies on the formula for expectation values as matrix elements of operators, which is itself derived from the Born rule.” In [8, p. 26] he questions, “If physical states, including observers and their instruments, evolve deterministically, where do the probabilities come from?" Again in his recent book [9, p. 131], Weinberg questions, “So if we regard the whole process of measurement as being governed by the equations of quantum mechanics, and these equations are perfectly deterministic, how do probabilities get into quantum mechanics?”
Maximilian Schlosshauer and Arthur Fine remark , “Certainly Zurek’s approach improves our understanding of the probabilistic character of quantum theory over that sort of proposal by at least one quantum leap.” However, they also criticize Zurek’s derivation of the Born’s rule of circularity, stating: “We cannot derive probabilities from a theory that does not already contain some probabilistic concept; at some stage, we need to “put probabilities in to get probabilities out.””
Maybe so, but I doubt it. The paper looks as if it reviews some standard QM theory, and shows that questions naturally have probabilities. Yes, sure QM has probabilities. It is when you make the leap to deterministic unitary theory and MWI that the probabilities disappear.
Weinberg is dead, so we cannot ask him if this paper solves the problem. I doubt that others are persuaded, but we shall see.
In the mean time, I cite this as proof that MWI currently has no way of saying that any outcome is more probable than any other. In other worlds, completely usuless as a scientific theory. Anyone who subscribes to it is a crackpot.
Unless this paper solves all the MWI problems. If the MWI advocates endorse this paper as a solution to their problems, then I will take another look at it. But that will not happen. They will just go on ignoring the fact that MWI cannot make any testable prediction.
Here is a podcast interview of Hugh Everett's biographer. He is described as having a hard life, and his MWI theory, which he preferred to call the "relative state", was not well appreciated in his lifetime. The interviewer, Steve Hsu is a believer.
They acknowledge that some journals refuse to publish anything in favor of MWI, and maybe half of physicists regard it as outlandish and ridiculous. But they also argue that it is essentially the same as decoherence theory, and that is very well accepted.
It is not the same. Decoherence is an attempt to understand how the wave function collapses, in the absence of an observer. Copenhagen followers regard it as a straightforward extension of known QM. MWI posits that decoherence is accompanied by a split in the universes, making many more.
Hsu says that the whole universe does not necessarily split; just the observer splits. Okay, but he really wants MWI for cosmology problems where there is no observer. The splits must be huge.
> "The author goes on to argue that he has solved these problems, and found a solution that has eluded physicists since 1926."
Actually not. I read through his paper (in its entirety).
He covers a lot of standard mainstream QM, and also attempts to relate the wavefunction of the mainstream QM to *some* notions from, or related to, QFT. He then argues that (a) since QFT fields are totally unpredictable, so must be the measurements in the mainstream QM, and (b) since virtual particles are always available to ensure that the equilibrium distribution of probabilities doesn't change [which he takes from Frank Wilczek], therefore, the probabilities associated with the different components of a QM wavefunction (split up in a suitable basis in which it 2+ components) don't change.
Interestingly, he also argues to the effect that since QFT fields are predetermined (he says, at least once, that they are "pre-ordained"), so must be the probabilities in QM. However, he never relates how is it that the QFT fields themselves come to have, simultaneously, the characteristics of being (a) pre-ordained, *and* (b) totally unpredictable.
[BTW, he doesn't touch on another characteristic of the QM measurements, viz., that they introduce irreversible changes, so let's not get into whether even the irreversibility has been pre-determined/pre-ordained, or not.]
All in all, he doesn't even begin solving the Measurement Problem.
But then, the point is, he *also* doesn't claim that he has solved the Measurement Problem. What he actually says, in conclusion, is this:
"We, therefore believe that a plausible answer has now been provided to the question, where do the probabilities in the measurement come from, thus removing one of the hurdles in resolving the century old measurement problem."
So, what he claims is a *plausible* removal of just *one* of the hurdles in solving the Measurement Problem. [And, he doesn't even pretend to have made a more comprehensive list of all such hurdles.]
In other words, the actual claim being made itself is a very, very, modest one.
As to the meat of the paper, viz., the discussion of the QM being related to (or, derived from, or, say, a higher-level view taken of) QFT: My point is, how is all this discussion useful, if nothing has been noted on how the pre-ordained QFT also comes to have probabilities.
I mean to say: People may say that the *real* reason why the deterministic Schroedinger evolution goes together with the unpredictable/probabilistic measurement events, is that QFT itself has these two aspects going together. But then, that's just shifting the burden of the Measurement Problem from the level of QM to the level of QFT. [Which makes it even more difficult to solve, not easier, if you ask me. And, which isn't really necessary, if you ask me. Taking the Problem to QFT adds a great deal of the further micro-scopic details, but does not eliminate the Problem itself; and the Problem already exists even at the much simpler level of even just the non-relativistic QM.]
Bottomline: No, the Measurement Problem hasn't been solved in this paper. And no, the author doesn't claim to have solved it. And no, he doesn't even begin tackling all the relevant aspects of the Measurement Problem in the first place --- not even in the context of QM, let alone in the context of QFT.
But, yes, what he says is better than MWI. Categorically better, in fact. [That's the [real] bottomline.]
PS: It's strange that despite spending so many years in California [which *is* in the USA], he still hasn't turned into a MWI supporter. That's strange. Quite strange, in fact.