The many-worlds interpretation of quantum mechanics has always been explained to me at a high level using examples of binary events (e.g. atom either did or did not decay at any given moment in time), which leads to a conceptually clean idea of "branching" into two distinct universes. But how does branching work if you have, say, a 70% probability of something happening (e.g. measuring the spin of an electron having gone through an SG apparatus oriented at an arbitrary angle)?
Do you say that 7 universes got to spin up and 3 that got spin down (to account for the 70% probability)? Does that mean those 7 universes are in-every-way identical copies of each other? But then what if you had something with a 71.87154819…% chance of happening? You would need an uncountably infinite number of branches to be able to represent arbitrarily precise probability ratios, and then subsets of those branches would contain uncountably infinite universes that are 100% degenerate and identical to each other. Is this what the standard many worlds interpretation assumes?
If not this, then what? You can't say that there's a 70% chance of universe A happening and a 30% chance of universe B happening if you're saying both happen. How does the many-worlds interpretation put a "weighting function" on different branches or outcomes?
Best Answer
Binary branching is just a simplification to make it easier to explain without math. The actual math is very simple, and can handle unequal probabilities.
At the simplest level, a branching occurs when you can write the wavefunction as a sum $$|\psi \rangle = |\psi_1 \rangle + |\psi_2 \rangle$$ where $|\psi_1 \rangle$ and $|\psi_2 \rangle$ are orthogonal and decohered, i.e. that there is no reasonable physical process that can make them overlap again. In this case we colloquially describe the two terms as "worlds" or "branches", and the probability of being in each one is the norm $\langle \psi_i | \psi_i \rangle$, which can be an arbitrary number between zero and one. The same logic goes for branching into more than two "worlds" at once, and repeated branching: you just get a sum of many terms, and the probability of each one is its norm.
After some comments, I get the feeling you really want a discussion of where the probability in the many worlds interpretation "comes from". Again, this is a very subjective and debatable thing, but my favorite take on it is "self-locating uncertainty".
Suppose that somebody kidnaps you, blindfolds you, and takes you somewhere in Uzbekistan. When you come to your senses, are you closer to Samarkand than Tashkent? You don't know for sure, so you can only answer in terms of probabilities. This is self-locating uncertainty: you're certainly in a definite place, and it's not like there are many copies of you running around, but there's probability nonetheless. You can use a variety of information to help. For example, if you weight by area, about 85% of the country is closer to Samarkand. (But this doesn't mean there are $85$ copies of you near Samarkand and $15$ copies of you near Tashkent!) But if you weight by population, substantially more of the population is closer to Tashkent, because it's the capital. Of course, which weighting is the correct choice depends on how the kidnappers set things up.
Now, suppose that after the spin of a particle is measured by a device, the state is $$|\psi \rangle = \sqrt{0.85} |\text{spin up measured} \rangle + \sqrt{0.15} |\text{spin down measured} \rangle.$$ You are living in one and only one branch of the wavefunction, but until you look at what the device is reading, you don't know which. At best, you can assign probabilities. The core assumption of many worlds is that the correct choice of probability (i.e. the choice that corresponds to what you actually observe, when averaged over many measurements) is to take the coefficient of each branch and take its norm squared, i.e. to assign an 85% chance to observing spin up.
If you ask where this assumption comes from, it's a perfectly legitimate question! However, the point is, there's no principle that says the probabilities have to be equal across branches. That's like saying every day must have a 50% chance of rain because it can either be rainy or not.