How hot is this chip? A silicon die typically has a few thousand compute tiles, each drawing power. Cooling control needs to calculate the whole steady-state temperature field before the die gets too hot. Figuring out how hot each part is, is equivalent to solving a large sparse linear system. Classical linear solvers are excellent tools but consume an unnecessarily large amount of power to provide an exact answer at scale. A fast approximation will often do just fine. Probabilistic solvers can tailor their accuracy to the computation budget. Distributed probabilistic solvers take that one step further and match computation itself to the sparsity pattern of the matrix. In this tutorial, we pose the problem of solving a sparse linear system of equations, and briefly discuss the classical Jacobi, Gauss-Seidel and conjugate gradient methods. We then present a probabilistic numerical solver as well as a distributed variant, based on a message passing algorithm. Solving the system becomes marginal inference in a Gaussian Markov random field, which is local, asynchronous, and comes with per-node uncertainty quantification. We provide Marimo (Python) and Pluto (Julia) notebooks for participants to explore and familiarize themselves with these concepts. We are looking forward to a fun interactive session.