= Solution
The <Born rule> assigns $p_i=\langle\psi|P_i|\psi\rangle$ to a <projective measurement> with <projection operators> $P_i$. Its numerical form can be shared by theories while the source of the <probabilities> differs. Comparing <Bohmian mechanics>, the <GRW model> and the <many-worlds interpretation> exposes three different roles: uncertainty over actual configurations, fundamental stochastic events, and weights for branch-relative expectation.
In <Bohmian mechanics>, the <Schrödinger equation> implies a <probability continuity equation> for $|\Psi|^2$ with current $j$, while the <guidance equation> transports an ensemble density $\rho$ with velocity $v=j/|\Psi|^2$:
$$
\partial_t|\Psi|^2+\nabla\cdot j=0,\qquad
\partial_t\rho+\nabla\cdot(\rho v)=0.
$$
Thus $\rho=|\Psi|^2$ is preserved: <quantum equilibrium equivariance>. If an apparatus interaction produces disjoint configuration regions $\Gamma_i$ with occupied-branch norm $|c_i|$, integrating the equilibrium distribution gives $\Pr(Q\in\Gamma_i)=|c_i|^2$. The actual outcome is fixed by initial configuration; the distribution describes the typical or uncertain configuration, not a fundamental random collapse. Equivariance alone does not prove equilibrium initially: every admissible density is transported by the velocity field. A typicality account privileges the equivariant $|\Psi|^2$ measure and then explains subsystem statistics conditionally; that explanatory premise must not be hidden in the continuity equation.<Quantum Equilibrium and the Role of Operators as Observables in Quantum Theory by Detlef Dürr and others>[analysis of measurement statistics from quantum equilibrium and configuration dynamics]{ref}
In the <GRW model>, the centre law $p(X)=\|L_i(X)\Psi\|^2$ is part of the stochastic dynamics. Since $\int L_i(X)^\dagger L_i(X)dX=I$, it is normalized. For two well-separated correlated pointer packets with coefficients $a,b$, integration over centres around their respective occupied regions gives approximately $|a|^2,|b|^2$; cross-contributions are suppressed at separations large compared with $r_C$. This explains why macroscopic records follow the <Born rule> approximately, without a special postulate applying only when an observer measures. However, the squared-norm centre law already carries the relevant mathematical probability structure. The theory relocates the rule into a universal physical law; it does not derive all probabilities from a deterministic equation. There are also small predicted deviations from standard unitary <quantum mechanics>.<Unified dynamics for microscopic and macroscopic systems by Giancarlo Ghirardi and others>[1986, normalized localization probabilities and macroscopic reduction]{ref}
In the <many-worlds interpretation>, every nonzero decohering component persists, so the question is not which single global branch will survive. It is why a rational observer should use branch weight $|c_i|^2$ for expectation or caring about successor records. Equal-amplitude symmetry and <decision theory> motivate equal weights for physically equivalent alternatives. For rational $|c_i|^2=m_i/M$, an ancillary refinement into $m_i$ equal-norm alternatives can motivate weight $m_i/M$, followed by continuity for other amplitudes. This argument requires assumptions that refining irrelevant details does not change value, that preferences respect the relevant quantum symmetries, and that weight tracks the intended notion of expectation. Those are substantive premises, not consequences of branch existence alone.<Quantum Probability and Decision Theory, Revisited by David Wallace>[rationality, equivalence and the scope of the decision-theoretic argument]{ref}
Naive branch counting is unsuitable: <decoherence> yields approximate, refinement-dependent branches, and splitting one record into more environmental records should not change its weight. Nor is diagonality of a <density matrix> already an account of probability. The comparison therefore has no common derivation from bare formalism. <Bohmian mechanics> needs a justified equilibrium or typicality measure, <GRW model> needs a justified stochastic law, and an Everettian account needs a justified connection from branch structure and rationality to experienced statistics. Each offers a coherent location for the <Born rule>; the remaining explanatory costs differ.
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