Does the universe split each time a qubit is measured ?

In the Many Worlds Interpretation of quantum mechanics it is thought that when a quantum wave function collapses to give one of the possible quantum outcomes, the universe splits into several universes. In each of those universes a different possible outcome is realized. For instance, assuming that a qubit is measured, its wave function collapses to give either up or down (1 or 0). In the Many Worlds Interpretation, the universe is split into two universes where in the first universe they observe 1 and in the second they observe 0. Check the wikipedia photo:



But, apart from that interpretation, each observation has a certain probability defined by Born's rule. That is, if we repeated the measurement (after re-initializing the quantum system to the same state) a very big number of times, then the fraction of times we observe a certain outcome is roughly equal to its probability.

So, if universe is split each time a qubit is measured into two universes, one where 1 is measured and another where 0 is measured, then how do we interpret probability of measurment ? It might be plausible to assume that the probability should hold relatively to each universe. For instance, for each of this universes, it should observe a probability consistent with the probability given by Born's rule. But since all paths are followed (correct me if I am wrong), then there exists exactly one universe (path) of all these branches (in the multiverse tree) where the measurement always took the same value. This was the obvious case for counter-example. It could also be noted that there would be a lot more universes where the probability isn't necessarily consistent the one given by Born's rule (the exact number -in the case of having two possible outcomes, could be given by $n! - \binom{n}{np}$ where $n$ is the number of measurements and $p$ is the probability of measuring one of the outcomes).

Another thing, if we assume that we could have an countably infinite number of measurements, then according the Many Worlds Interpretation, we will end up with an uncountable infinite number of universes.