Will your luck turn?

May 31, 2018 By Alan Salzberg

In the third quarter of the NBA Eastern Conference Finals last week, Boston missed all 14 3-pointers they attempted.  This led the announcer to say something like, "by the law of averages, they are bound to improve." 

The law of averages is the flawed idea that if you have a string of bad luck then it will be balanced by a string of good luck.  "If I lose a coin toss 10 times in a row, then surely I will win on the next toss," says the law of averages.  However, this doesn't hold because the outcome of each toss is independent of all previous tosses. 

The law of averages is different than 'regression to the mean.'  Regression to the mean is the tendency of a random process to return to the mean.  If I have a string of 10 heads, it is likely that the next 10 will be about 5 heads and 5 tails.  Similarly, if I have a string of 10 tails.  (It's also true if I get 5 heads and 5 tails in 10 tosses).  Regression to the mean is really no more than the effect of the fact that random good or bad luck of the past is not predictive of the future.  In his book Thinking, Fast and Slow, Nobel laureate Daniel Kahneman describes the experience of an air force captain who says only negative reinforcement works.  How does the captain know this?  He praises his pilots after a great test exercise and they nearly always do worse the next time.  He criticizes them after a particularly bad mission and they nearly always do better.  What the captain missed is that he is witnessing the phenomenon of regression to the mean.  The little luck that led to the particularly good or bad test exercise returns to its long run average, leading to an average mission the next time, after both the great and terrible test exercises.

So, if Boston's 3-point shooting last week was really independent of the Cav's defense that night, Chris Paul's average, or anything else, then, simply due to the regression to the mean, they would be likely to shoot better in the fourth quarter than the third.  And they did—they shot 1 for 7 instead of 0 for 14, and lost the game and the finals to the Cavs.