Home > Uncategorized > Playing with Percentages — Red Cards

Playing with Percentages — Red Cards

Knowing just enough about statistics and probability to make me dangerous, the posts here generally shy away from too much analysis.  They state the facts we found, any thoughts we have about them, and then we leave the rest to the readers, generally speaking.  But, it’s fun to dabble with them, so here we go:

Assumptions:

  1. Referees attempt to make the correct call in every match they officiate;
  2. For the purpose of evaluating any referee discrepancies between club and national team matches, full national team matches (i.e., ones that do not impose an age limit, e.g. U-20, the Olympics, etc.) are all the same (regardless of whether a national team includes all of its starters or is mostly backups in a given game.)
  3. Teams approach tournament matches differently than they do friendlies or even qualifiers;
  4. Full-squad tournaments (even if any particular team doesn’t bring its ‘A’ squad) are more indicative of a team’s tendencies than are youth tournaments;
  5. Red cards are independent events (I know this isn’t quite true, but it simplifies the analysis… I’d welcome the efforts of someone who can better account for them.)
  6. The statistics on worldreferee.com are accurate.

Coming into this tournament, the U.S. had played 19 tournament matches in the last five years (2007 Gold Cup, 2007 Copa America, 2006 World Cup, 2005 Gold Cup).  In those matches, it was red carded in three matches (four total cards) and its opponents in seven matches (seven total cards).

So, let’s say that:

  1. In any specific tournament match, the U.S. has a 3/19 chance (16%) chance of receiving at least one red card.
  2. For any given red card issued during a U.S. match, there is a 4/11 (36%) chance that it will be issued to the U.S.

Now let’s look at the officials for the U.S.’s first four matches at the Confederations Cup 2009, as they were prior to the start of the tournament

  1. Italy — Pablo Pozo (CHI).  Pozo has issued red cards in 17 of the 36 matches he has officiated (47%).  In matches with full national teams, he has issued red cards in 3 of the 3 matches he has officiated (100%).
  2. Brazil — Massimo Busacca (SUI).  Busacca has issued red cards in 7 of the 78 matches he has officiated (9%).  In matches with full national teams, he has issued red cards in 2 of the 23 matches he has officiated (9%).
  3. Egypt — Michael Hester (NZL).  Hester has issued red cards in 4 of the 13 matches he has officiated (31%).  In 7 national team matches, he has issued red cards in 3 of those matches (43%).
  4. Spain — Jorge Larrionda (URU).  Larrionda has issued red cards in 51 of the 109 matches he has officiated (47%).  In matches with full national teams, he has issued red cards in 9 of the 26 matches he has officiated (35%).

Now let’s look at the teams’ statistics coming into the tournament:

  1. Italy —   In the last five years, Italy has been red carded in two of its 17 tournament matches (12%) (2 cards).  Its opponents have been red carded in four of those matches (24%) (5 cards; One game, US-Italy World Cup, had 3 total cards.).  5 of 17 matches have had at least one red card.  (29%)  Italy was last red carded on April 1, 2009 in a draw with Ireland during a World Cup qualifier.
  2. Brazil — In the last five years, Brazil has played in 22 tournament matches.  It was not red carded in those matches;  its opponents were red carded in two matches (9%).  Brazil was last red carded on June 6, 2009 in a 4-0 victory over Uruguay in World Cup qualifying.
  3. Egypt —  In the last five years, Egypt had not been red carded in an international tournament.  Its opponents have been red carded in 2 of its 15 tournament matches (13%).  Prior to the tournament, Egypt was last red carded on July 29, 1999.
  4. Spain —  In the last five years, Spain has not been red carded in an international tournament.  Its opponents have been red carded in 2 of its 13 tournament matches (15%).  Spain was last red carded on September 8, 2007 during a Euro Qualifier draw against Iceland.

Let’s look at the probability of the US getting red carded in three of four matches a couple ways:

  • The US has a 16% chance of a red card in any given match and an 84% chance of not getting one.  So 16% x 16% x 84% x 16% = 0.3% chance of reds in three of four matches.
  • The probability that in four matches, one refereed by Pozo, Busacca, Hester, and Larrionda, that there would be red cards in all but Hester’s match:  47% (Pozo) x 9% (Busacca) x 69% (Hester) x 47% (Larrionda) = 1.4%.
  • Same as above, but using full national chances rather than referees’ overall chances:  100% (Pozo) x 9% (Busacca) x 57% (Hester) x 35% (Larrionda) = 1.8%.
  • The probability that in four matches, one refereed by Pozo, Busacca, Hester, and Larrionda, that there would be red cards issued in three matches:  9%  (We’re not showing our work here, but we’re adding the probabilities for the four possible scenarios in which this is true, e.g., if Pozo didn’t show red but the others did; if Busacca didn’t show red but the others did, etc.)
  • Same as above, but using full national chances rather than the referees’ overall chances:  18%.
  • Chance that in four matches, Italy, Brazil, and Spain’s opponent would be red carded but Egypt’s would not: 24% x 9% x 87% x 15% = 0.3%
  • Chance that in four matches, at least one red card would be issued in three of four separate matches involving Italy, Brazil, Spain, and Egypt:  1.3%  (Same as with the officials, we’re not showing our work here, but are doing a similar thing.  We are using 29% for the chance of a red card during an Italy match to account for the possibility that Italy and not its opponent would be red carded.  Because the other teams don’t have tournament red cards, the chance of a red card in their match is equal to the chance of an opponent’s red card.)

Ultimately, what has happened to the United States at this Confederations Cup is beyond reasonable predictions.  At most, we could expect that only 18% of the time would four matches handled by these officials include three with at least one red card.  Let alone they would all be (a) to the same team; (b) to the same position on the same team; or (c) straight reds.  And that’s ignoring the fact that only one of our opponents (Italy) sees red cards more than 15% of its tournament matches in the past 5 years.

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