Friday, March 19, 2010

NCAA Hoops Day 2 -- Georgia Tech FT Shooting

The Georgia Tech Yellowjackets (also known as the Ramblin' Wreck) finished the regular season ranked 12th (and last) in team free-throw shooting in Atlantic Coast Conference (ACC) men's play at .645 (here and here). One of the great things about NCAA tournament action is that odd things can -- and often do -- happen. That includes Georgia Tech making 24 out of 25 free-throw attempts tonight in a 64-59 win over Oklahoma State.

It is a familiar probability question, known as the binomial expansion, to ask how likely it is for a process with a long-term baserate success probability of [fill in the blank] to succeed in a new set of attempts [blank] times or more out of [blank] attempts. In Georgia Tech's case, we would ask how likely it is for a team with a .645 baserate to make 24 or more free-throws out of 25 (assuming the shots are independent of each other, such as coin tosses). There's even an online calculator for the occasion that, when supplied with the particulars of Georgia Tech's accomplishment, yields a probability of .00026 (26-in-100,000 or approximately 1-in-4,000) of the Yellowjackets doing what they did.

Where does that final probability come from? First, we calculate the probability of Georgia Tech going a perfect 25-of-25. If you think of the likelihood of rolling double-sixes with a pair of dice, it's the probability of a six on a single die, 1/6, raised to the second power (given there are two dice) or 1/36. Analogously in our basketball example, we take the Yellowjackets' probability of success on a single free-throw attempt, .645, to the 25th power, yielding .000017 as the probability of a perfect run of 25 out of 25 made free-throws.

We also have to calculate the probability of 24 made free throws and one miss, which is: (.645 to the 24th power) times .355 (the latter being the probability of a miss). That yields .0000095. However, there are 25 different sequences in which a team could make 24 free-throws and miss one; the one miss could occur on the first, second, third,... all the way through the 25th attempt. We must therefore multiply .0000095 times 25, yielding .0002375.

Lastly, we add .000017 (probability of making all 25 attempts) to .0002375 (probability of making 24 out of 25, taking into account all the different possible sequences of doing so), to get .0002545 (within rounding error of the .00026 from earlier).

A few cautions are in order. First, with all the tens of thousands of games taking place in a single season at all levels of college and professional basketball -- never mind multiple years of play -- a feat like Georgia Tech's will occur every so often. Second, I did not take a random cross-section of games, but rather jumped straight to Georgia Tech vs. Oklahoma State because there was a dramatic accomplishment. Third, it could have been the case that the Yellowjackets had one or two great free-throw shooters (along with a bunch of poor shooters who bring down the average) and that these stellar shooters took a disproportionate share of free-throw attempts against Oklahoma State. As it turns out, however, none of the eight Yellowjackets who played tonight is any better than a .787 free-throw shooter, as seen in the team's statistics.

1 comment:

Movement2 said...

A classic case of "rare things are expected to happen but only rarely.