The Thunder and Warriors played a very entertaining game last night. All the stars showed up (Monta Ellis, career high 48 points; Kevin Durant, 33 points, game winning shot, close to a triple double). The defense wasn’t terrible — the Warriors especially made a bunch of tough shots. Close games like this are generally decided by luck, but there were two interesting decision points in the endgame where each team affected the odds:

**Down one, should you shoot early or late?**

With about 22 seconds left, down one point, the Thunder had Durant drive right to the hoop and go for a quick shot — air ball, but the Warriors knocked it out of bounds. On the next inbounds play, Durant pulled up immediately and banked in a (relatively) open jumper to take the lead with 16 seconds remaining. This gave the Warriors plenty of time for a rebuttal, and the Warriors announcers were confused that the Thunder didn’t run down the clock to take the last shot.

The Thunder clearly wanted to shoot quickly. Did this help or hurt their chances of winning? It gives the Warriors another chance if they make, but it also allows the Thunder a second chance if they miss. I can work through the probabilities pretty easily since there wasn’t much time left, but you can skip below for the answer if you like. Here are the parameters (assuming no more than three possessions total for simplicity, neither team takes a three unless they need it for a tie, and subsuming turnovers and offensive rebounds into the overall shooting percentages):

- Prob(Thunder make shot if taken early) = p
- Prob(Thunder make shot if taken late) = q
- Prob(Thunder make shot given lots of time next possession) = r
- Prob(Thunder make shot given little time next possession) = s
- Prob(Thunder make a 3 given lots of time next possession) = t
- Prob(Warriors make a free throw) = f
- Prob(Warriors make shot given lots of time) = x
- Prob(Thunder win in OT) = 50%

If the Thunder take the shot early, their probability of making the shot and winning is

p(1-x) + pxs

= p(1 – x + xs)

The first term is the probability that the Thunder make and the Warriors miss (which is what actually happened). The second is the probability that both teams make but the Thunder make again for the win. The amount of time left for the Thunder on their second possession depends on the Warriors’ strategy, of course. They probably won’t have much time (maybe no time at all), so s is likely quite small.

Their probability of missing the early shot and winning is a little more complicated, since it involves a variety of free throw outcomes:

(1-p)(1-f)(1-f)r + 0.5(1-p)2f(1-f)r + 0.5(1-p)fft

= (1-p)(r(1-f)^2 + rf(1-f) + 0.5tf^2)

The first term is the probability that the Thunder miss, the Warriors miss both free throws, and then the Thunder make. The second is the probability that the Thunder miss, the Warriors make one free throw, and the Thunder tie and win in overtime. The third is the probability that the Thunder miss, the Warriors make both free throws, the Thunder tie (with a three) and win in overtime.

The total probability of a Thunder win, given an early shot, is the sum of these two:

p(1 – x + xs) + (1-p)(r(1-f)^2 + rf(1-f) + 0.5tf^2)

Now, what if the Thunder follow the Warriors announcers’ advice and run the clock all the way down? This one is easy. The probability of a win is just the probability of a Thunder make: q. I could allow the Warriors a rebuttal with one or two seconds left, but just like with s above, this probability will be quite small and won’t add much.

Let’s try some reasonable parameters:

- p = 0.5
- q = 0.5
- r = 0.5
- s = 0
- t = 0.2
- f = 0.8
- x = 0.5

Plugging in these parameter values says that the Thunder will win just 33.2% of the time if they shoot early, and 50% of the time if they shoot late. The problem is, even when the Thunder make the shot (50% of the time), they still lose the game half the time. The Warriors are a strong free throw shooting team, so the ability to put them on the line after a miss doesn’t help much.

What parameters might be off? Well, Monta Ellis ended up taking a tough, rushed three pointer after Durant made his shot (he had a career high 48 points in the game and fell into the trap of taking more difficult shots once he “caught fire”). Also, shooting early (p) could be higher percentage than shooting late (q), since the defense doesn’t know exactly when the shot will happen. If I decrease x to 40% (since the Warriors didn’t have a ton of time), increase p to 55%, and decrease q to 40%, then shooting early wins 40.4% of the time, leaving the two strategies about equal. Against a strong offense that makes it’s free throws, it looks like taking the shot early only makes sense if you get a really good look at the hoop.

**When to throw the ball skyward?**

With 3.5 seconds left and a one point lead, the Thunder still had to inbound the ball. Westbrook received the pass near half court, waited for the foul, and sank his free throws. Brandon Rush missed a desperation three for the Warriors, but what if Westbrook had tried the Rubio chuck to the ceiling to kill the clock? After I wrote about Rubio’s play, commenter Counterpoint was quick to remind me when this failed: Morris Peterson caught a similar chuck with just enough time to sink the winning shot. Let’s think about when the chuck works and when it doesn’t.

The main piece of the puzzle is how long the ball stays in the air. For this, I turned to my physics expert, Brother Evan. We calculated the amount of time it takes for a ball to drop from 20 feet (6.1 meters) with the following standard acceleration equation (h is the height of the ball, g = 9.8 m/s^2 is gravitational acceleration, and t is time to reach the ground):

h = 1/2gt^2, or

t = sqrt(2h/g)

=> t = sqrt(2*6.1/9.8) = 1.1 seconds

A ball thrown to 20 feet will take exactly the same amount of time to reach its peak as it will to fall from its peak (the paths traveled up and down are symmetric), so the total time in the air for a ball thrown 20 feet is about 2.2 seconds. We can also calculate the starting and finishing velocity of the ball quite easily (v = gt = (9.8)(1.1) = 10.8 m/s = 24.1 mph), though that’s not too important for our purposes.

So, in Westbrook’s case, chucking the ball wouldn’t have killed the entire 3.5 seconds. Even though they were in the offensive half of the court, the Warriors had a time out, so they could have quickly called it and gotten up a desperation shot. He made the right decision by holding the ball and waiting for the foul.

What was different for Rubio? He had a similar amount of time to kill, but he was in his own end collecting a rebound, with everyone else in the same end. Not only did he waste 2 to 2.5 seconds by chucking the ball, but he threw it long down the court, away from every player, and away from the basket where the Nets needed to score. These factors combined to give the chuck a 100% success probability. The Morris Peterson play differed in that there were players all over the court, so there was nowhere safe to throw the ball.