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In figure skating, past controversy over judging has led to new rules that invoke a random selection of judges. That randomness can, and in fact has, affected the outcome of competitions.
The medals awarded in the Pairs competition of the 2006 World Figure Skating Championships did not reflect the consensus of the judges. Why? Under the International Skating Union (ISU) rules, chance (not CHANCE) played a role in scoring the competition. The scores of three judges were discarded at random in the Short Program, and another set of three judges' scores were eliminated in the Free Skate.
Unfortunately for Russian bronze medalists Maria Petrova and Alexei Tikhonov, the randomness introduced by the ISU rules (not simply their performances on the ice) made the difference between silver and bronze. In fact, the competition was close enough that different panels of judges could easily have awarded them gold. I can't give Petrova and Tikhonov the silver medal I think they deserve, or the gold they might have won with a bit of luck. However, I can say that the consensus of 12 judges would have awarded the silver medal to Petrova and Tikhonov at the 2006 World Figure Skating Championships.