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.

中文翻译：

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足球小组抽签概率和修正

本文考虑了设计足球小组抽签机制的挑战，该机制对所有有效的抽签分配有统一的分配，但也具有娱乐性、实用性和透明性。尽管这个问题在完全对称的问题中是微不足道的，但当每个竞争团队之间存在不可交换的抽签约束时，它就变得具有挑战性，从而导致对称性被打破。我们解释了如何模拟 FIFA 顺序抽取方法并通过与均匀拒绝采样器进行比较来计算其抽取的不均匀性。然后，我们提出了几种实现均匀分布的实用方法，同时仍然以适合电视转播的方式使用球和碗。解决方案也可以交互地进行。我们提供的通用方法可以很容易地应用于不同的比赛抽签，并且不限于足球赛事。