Football group draw probabilities and corrections

This article considers the challenge of designing football group draw mechanisms, which have a uniform distribution over all valid draw assignments, but are also entertaining, practical and transparent. Although this problem is trivial in completely symmetric problems, it becomes challenging when there are draw constraints that are not exchangeable across each of the competing teams, so that symmetry breaks down. We explain how to simulate the FIFA sequential draw method and compute the nonuniformity of its draws by comparison with a uniform rejection sampler. We then propose several practical methods of achieving the uniform distribution while still using balls and bowls in a way which is suitable for a televised draw. The solutions can also be carried out interactively. The general methodology we provide can readily be transported to different competition draws and is not restricted to football events.