World Cup draws and probabilities

How do constraints affect the chance teams will face each other?

The televised World Cup draw selected groups for the men’s football competition. There are three good properties for a drawing system to have:

• Fairness: no team should have greater chances than others of a ‘tough’ group.
• Balance: the groups should be at a similar competitive level.
• Uniformity: the probability of each valid outcome should be the same.