Football (“soccer”) is the most popular sport in the world, particularly in Europe, South-America, Africa and Asia. The most watched tournament is the UEFA Champions League, UEFA being the Union of European Football Associations. The UEFA Champions League is also the most-revenue generating tournament in football. Professional football has been troubled by multiple scandals in the past few years, including accusations of corruption and match-fixing.
Recently, the UEFA fell again under a cloud of suspicion. The 2013 Champions League draw ceremony for the quarter-finals resulted in the following four matches:
- Málaga—Borussia Dortmund
- Real Madrid—Galatasaray
- Paris Saint Germain—Barcelona
- Bayern München—Juventus
The outcome of the quarter-finals draw led to heated discussions in European sports programs on television and radio. Several sport journalists accused the UEFA of manipulation in order to make possible the commercially most interesting semi-finals and final. The most explicit accusations came from a former international soccer referee (details here).
It is quite remarkable that the Big Four of the eight teams avoided each other in the quarter finals. The Big Four are the two Spanish teams Barcelona and Real Madrid and the two German teams Bayern München and Borussia Dortmund. Moreover, it is remarkable that none of the Spanish teams was paired with the third Spanish team Málaga, an unattractive opponent for both Barcelona and Real Madrid. The quarter-finals draw kept open the possibility of a dream final between Barcelona and Real Madrid. What are the chances that this particular quarter-finals draw is rigged?
To answer this question, let us first calculate the probability that the Big Four avoid each other and neither Barcelona nor Real Madrid plays against Málaga when it is assumed that the eight teams are paired randomly. Under random pairing the Big Four avoid each other with probability
Since the draw ceremony involves eight teams, Málaga must be paired with one of the teams from the Big Four if the Big Four avoid each other. Hence, under the assumption of random pairing of the teams, the probability that the Big Four avoid each other and neither Barcelona nor Real Madrid is paired with Málaga is given by . This probability is small but not exceptionally small and therefore frequentists may argue that the result of the quarter-finals draw is no surprise when taking into account that there are many soccer tournament draw ceremonies over the years.
However, this is bad reasoning. The discussion is not about many tournament draw ceremonies, but about a particular soccer tournament ceremony for which there is reasonable ground to believe beforehand that the draw ceremony could be manipulated. In this situation it is appropriate to use the Bayesian approach. Bayesian analysis requires that before the draw ceremony takes place you quantify your personal belief that the draw ceremony will be manipulated.
Suppose you believe that the prior probability of a manipulated draw ceremony is at least 20%. Many soccer fans will consider a prior probability of 20% for a manipulated draw as a conservative estimate. By the formula of Bayes, your personal belief of a manipulated draw after hearing the result of the draw is given by a posterior probability of at least 68.6% if your prior probability is at least 20%. The easiest way to calculate the posterior probability is to use Bayes formula in odds form:
In our example the hypothesis H is the event that the draw ceremony is manipulated so that the Big Four avoid each other and neither Barcelona nor Real Madrid is paired with Málaga, is the complement event that the teams are paired at random, and the evidence E is the event that the Big Four avoid each other and neither Barcelona nor Real Madrid is paired with Málaga. If your prior probability that the draw ceremony will be manipulated is r%, then the prior odds is and the likelihood ratio is . This gives the posterior odds
Since , it follows that your posterior probability of a manipulated draw ceremony is given by
This posterior probability is equal to 0.6863 if your prior probability is represented by r=20%. The Bayesian analysis show that the suspicion voiced in the sports programs after the announcement of the result of the Champions League quarter-finals draw is certainly not unwarranted. Incidentally, in the end the final was not played between the two Spanish teams Barcelona and Real Madrid but between the two German teams Bayern München and Borussia Dortmund with Bayern München as winner.
This post emphasizes once again the importance of Bayesian thinking which is an indispensable part of statistical reasoning. Bayesian thinking is advocated in my book Understanding Probability (Cambridge University Press, third edition, 2012). This feature distinguishes my book from other introductory probability books and was praised in this book review.
I dare to say that the leading textbooks for introductory probability courses badly fail in the attention paid to the Bayesian approach. Students should be better trained to think in the Bayesian way. Every modern course on introductory probability should give greater recognition to the probabilistic ideas of Bayesian thinking and show that Bayes’ rule is the rational basis for answering probabilistic questions from real life.