In the middle of a fascinating article on South Africa’s preparations for the World Cup, R. W. Johnson makes the following offhand remark:
Any minute now the usual groaning will be heard from teams which claim that they, uniquely, have been drawn in a ‘group of death’. What is the point, one might ask, in groaning about a random draw? Well, the trouble starts there, for the draw is not entirely random. In practice, seven teams are seeded, according to how well they’ve been doing in international matches, along with an eighth team, the host nation, whose passage into the second round is thus made easier – on paper. The draw depends on which balls rise to the top of the jar and thus get plucked out first; but it’s rumoured that certain balls get heated in an oven before a draw, thus guaranteeing that they will bubble to the top. The weakest two teams aside from South Africa and North Korea are South Korea and New Zealand. The odds are, of course, heavily against any two or more of these bottom four finding themselves in the same group. If they do, we will have to be deeply suspicious of the draw.
This got me wondering. What is the probability that the bottom four teams will actually end up in different groups?
Given the rules as stated above, eight of the teams (including South Africa) start in eight different groups. There are 24 slots remaining. Now let’s assign the next three low-ranking teams. The first has a 21/24 chance of being in one of the seven groups that does not have South Africa; the next has a 18/23 chance of being in one of the six remaining groups, and the next has a 15/22 chance of being in one of the five remaining. Combining these, the probability that the bottom four teams are in four different groups is 1-(21/24)*(18/23)*(15/22) = 0.53. (Unless I did the calculation wrong. Such things happen.)
So, no, I don’t think that if two of these teams happen to find themselves in the same group, that “we will have to be deeply suspicious of the draw.”
P.S. The 53% event happened: the four bottom-ranked teams are in different brackets. So we can breathe a sigh of relief.