Football Group Draw Probabilities and Corrections

This paper considers the challenge of designing football group draw mechanisms which have the uniform distribution over all valid draw assignments, but are also entertaining, practical, and transparent. We explain how to simulate the FIFA Sequential Draw method, to compute the non-uniformity of its draws by comparison to a uniform Rejection Sampler. We then propose two 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 tried interactively.