##### Rebounding, Part II: Defensive Rebounds

Part I is here. While each offensive rebound is worth The situation is different on defensive rebounds. Anyone who’s watched basketball can remember many situations where the defensive team surrounded the ball with everyone boxed out. Then, which player happened to get rebound is irrelevant, as discussed above. So a defensive rebound is not as valuable as an offensive rebound, since it’s more common, and because if a particular player didn’t get the defensive rebound, there’s still a good chance someone else on his team would.

So, I’m going to use the idea of removing the player who got the defensive rebound from the play. The value of the rebound is determined by how likely it would be for the opposing team to get the offensive rebound if he didn’t. But how to quantify this? I’m going to consider the rebound as a loose ball, with each player in the region having an equal chance of getting the rebound. Assume one offensive rebounder, with a chance (from the league average) of 28.5% of getting the rebounder. Now, if each rebounder has an equal chance of getting the ball, that means there have to be 2.5 defensive rebounders (on average, obviously) in the vicinity, each with a 28.5% chance of getting the board, in order to give the total defensive rebounding probability of 71.5%.

Now, what happens when you take away one of the defensive rebounders, when you remove his chance of getting the rebound. Then, it’s only 1.5 to 1 in favor of the defense, and the chance of the offensive rebound is 40%. So, if you remove the defensive player who actually got the rebound from the play, the offense then has a 40% chance of getting the board. So the value of the defensive rebound is 40% of a possession, or 0.4 points.

This might seem a little arbitrary, but it ends up giving a result which is interesting and, on further consideration, very plausible for a completely independent reason. Let’s use these numbers to figure out what the value is of a player who gets a typical 10 rebounds per game. On average, he would be getting 7.15 defensive rebounds and 2.85 offensive rebounds. The value of the offensive rebounds is easy, 2.85 points. The value of the defensive rebounds is 7.15 * 0.4 which come out to, surprise, 2.85 points. This means that, with this formula, the average team gets the exact same value from its offensive rebounds as it does from its defensive rebounds.

And it wasn’t just a lucky coincidence, either. If you go back and do the algebra (which is left as an exercise for the reader), it’s straightforward to show that the methodology I used above to calculate the value of defensive rebounds will, no matter what the base probabilities for offensive and defensive rebounds, produce the result that the average team (and the average player) gets the exact same value from their offensive as from their defensive rebounds.

Now, in physics, these sorts of strange coincidences come up all the time—you work through lots of seemingly aimless math and end up with a very simple or symmetric result. And when you do that, it’s usually a sign that your theory is correct. Now, I don’t know if God plays dice with basketball or not, but this nice symmetric result gives me more confidence in my methodology. And in the next post, I’ll explain why I think this result makes sense.