Today's Math Lesson: Monty Python and the Holy Hot Hand

I know that you folks probably aren't reading too many math blogs, so I feel it is my duty to help out in a small way. Today's lesson:


From Wikipedia:  In probability theory, the law of large numbers is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.    

Stated a little differently, if a fair coin is tossed a large number of times, you would expect about 50% heads and 50% tails.  You have to be careful how you apply it though. If after 1000 tosses the difference between heads and tails is 20, on the next flip the difference is just as likely to go to 21 as it is to 19. We aren't saying the difference gets smaller. But the ratio between the difference and the number of tosses should approach zero (if the difference grows, it grows more slowly than the number of tosses grows). Being 20 apart after 1000 tosses is a more significant difference than being 25 apart after 2000 tosses.   

So what does this have to do with today's math lesson? Well, having just watched the Chicago Bulls destroy the Atlanta Hawks to go up 2 - 1 in the conference semi-final (believe me, this series is over) and watching Derrick Rose get 44, the age old question of the "hot hand" came up again. Luckily, mathematics is there to triumph over perception.     

Here's where the law of large numbers comes in. Sometimes, people who believe in the law of large numbers, and that's most of us, including mathematicians, come to think there is something called the law of small numbers that functions much the same. That is, after 10 tosses, the results should be roughly 5 heads and 5 tails. Unfortunately, that is not true. And here's where people get off track. They see a streak in basketball, where Dirk Nowitzki hits three in a row, and say, "He's got the hot hand. Feed him the ball every time."  And what happens when you do that is that shooters get worse than normal.     

That's what John Huizenga of the University of Chicago and Sandy Weil of Sportsmetricians Consulting Group found after analyzing every shot from the 2002-03 season to 2007-08. They highlighted 49 players that could be termed high level jump shooters in the NBA (these are the guys everyone would think of when saying someone has a hot hand). Here are their findings about these players:    

i) after making a jump shot, players tend to take harder shots, lowering their field goal shooting by 3.5%,    

ii) after making a jump shot, players shoot 16% sooner than after a missed jump shot,   

iii) after making a jump shot, players shoot their team's next shot 34% of the time, as opposed to 25% of the time after a missed jump shot,  

and, my favorite,
iv) if everyone on the team behaves this way, it will cost the average team 4.5 wins a season.   

One of my favorite writers about statistics and public perception, Amos Tversky of Stanford University, was involved in a study published in 1985 that looked at Philadelphia 76er games and found:  

"the outcomes of both field goal and free throw attempts were largely independent of the outcome of the previous attempt. Moreover, the frequency of streaks in players’ records did not exceed the frequency predicted by a binomial model that assumes a constant hit rate."   

That is, when we look at several strings of five shots, where the shooter has a shooting percentage of 40% (the expected result is 2 made out of 5 shots), there should be some strings where they make five in a row, just as there should be some strings where they miss five in a row (Kyle Korver: 1 of 9 in Wednesday's game). The number of times each happens is consistent with what mathematics tells us it should be.  That is, the number of streaks in shooting in basketball is just what random chance tells us it should be. 

Pretty much every study for 20 years has told us the same thing. There is no such thing as a hot hand in basketball. Which brings us to the video of the day (with a tip of the cap to ESPN TrueHoop blog), wherein the same sort of vague science is used to determine whether a woman is a witch in Monty Python and the Holy Grail. One of the best pseudoscience vignettes I've ever seen - even though it works out right in the end. 

The best line - 

A normal looking man:  "She turned me into a newt......... I got better."