The prediction of knockout tournaments represents an area of large public interest and active academic as well as industrial research. Here, we show how one can leverage the computational analogies between calculating the phylogenetic likelihood score used in the area of molecular evolution to efficiently calculate, instead of approximate via simulations, the exact per-team tournament win probabilities, given a pairwise win probability matrix between all teams. We implement and make available our method as open-source code and show that it is two orders of magnitude faster than simulations and two or more orders of magnitude faster than calculating the exact per-team win probabilities naïvely, without taking into account the substantial computational savings induced by the tournament tree structure. Furthermore, we showcase novel prediction approaches that now become feasible due to this order of magnitude improvement in calculating tournament win probabilities. We demonstrate how to quantify prediction uncertainty by calculating 100,000 distinct tournament win probabilities for a tournament with 16 teams under slight variations of a reasonable pairwise win probability matrix within one minute on a standard laptop. We also conduct an analogous analysis for a tournament with 64 teams.Supplementary InformationThe online version contains supplementary material available at 10.1007/s11222-023-10246-y.