# Michigan foul rate & autobenching

First diary post!

Lets take a look at our foul rates in light of the autobenching discussion. More than trying to take a position on the debate I wanted to add some #'s similar to what was requested.

Some basics: Foul rate per 40 is the number of fouls that the player would get on average if they played all 40 minutes of the game.

This is the actual distribution of foul counts per game on the season so far. We have 2 foul outs on the season: Teske @ Iowa & Matthews @ Penn State.

The same table but looking at the percentages:

OK so lets mix in some statistics here. I use what's called a poisson distribution to estimate the expected counts of the number of fouls based purely on the average foul rate.

Note: This makes some assumptions/simplifications that aren't too terrible (beyond the scope for me to explain here). Also, I'm not doing any complicated modeling taking into account any other variables like home/away, change in behavior of opponent/player, opponent foul drawing rate, etc.

So here are the expected counts of fouls for each player based on the number of minutes they play per game and their respective foul rate:

So based purely on probability we would expect each player to foul out at the percentages highlighted in yellow.

Lets compare what we would statistically expect based on their foul rate to the actual distribution of fouls:

A few things jump out to me personally here:

- The general squeezing towards the middle of the distribution (aka red at 0 & 5 and green at 2 & 3). This shows that the players/coaches are controlling their rates either through substitution or altering their play to be more/less aggressive based on # of fouls (duh).
- Some players control foul rate better than others. E.g. Livers pretty much doesn't alter his foul rate (either doesn't have to or can't). Matthews seems to be excellent at controlling his number of fouls -- he lands right at 2 much more often than we'd expect and his 0 and 4+ are reduced. This also begs the question what happened at Penn State?? I didn't see the game(flight) but 5 fouls for him is some kind of crazy outlier given his control/foul rate. Matthews hadn't previously/since reached even 4 fouls in a game.
- Without some sort of benching/play alteration we would expect to see a significant # of foul outs ~ 5-7% per player & 4 fouls ~10% of the time. We've seen 2 total foul outs. So autobenching is certainly working to limit foul outs.

For the statsy inclined people, here are some tables for comparing expected foul probabilities for different minutes played.

Example of how you could read the charts (see below):

So lets say we were planning on playing Simpson all 40 minutes (hook shots galore!). We'd expect (disregarding any ability to control fouls by adjusting play as mentioned above) an 11% foul out chance. Now lets say he immediately gets 1 foul within the first minute. Now his foul-out probability jumps to 13%+11%= 24% because he only has 4 left, and a 21+13+11=45% chance to get at least 4 fouls, and so on.

We decide to take a risk and leave him in. He plays 5 minutes with no fouls so his % is now down to 18%. At the 30 minute mark he picks up another one. Now he's got 2 fouls and a 17+8+4=29% chance of fouling out if he plays every minute the rest of the game. Ouch. Lets sit him since he probably needs a breather anyways.

Now do we wait for 5 minutes or sit him till halftime? If he goes back in at 25mins: 13+5+2=20% vs 20mins: 9+3+1=13%. Up to you.

Picking up a 3rd foul right after half: 22+9+3+1=35% is definitely worth a sit.

4th foul at the 10 minute mark: 33+10+2=45% obviously also worth a sit.

Playing out last 5 minutes with 4 fouls: 25% chance to foul out.

Here is a better visualize of the risk at different time-points vs foul counts:

Obviously all this has to be taken alongside a vast combination of other game factors like when players would naturally be getting rest, player matchups/combinations, in-game momentum and the score, needing Simpson pg skills for the end of the game, etc, which Beilein knows best.

Conventional autobenching seems to do an ok job of getting players away from the dangerzones above the high 20s.

There's probably some excessive benching on every team (very hard to measure) -- I'd guess specifically early in games and with players who have low foul rates and a strong ability to control their foul rate. You're probably fine playing Matthews with 2 fouls in the last 5 minutes before half or after an immediate foul #1 in the first 5 minutes.

Ideally, you'd like to have a sort of smooth decision making curve which keeps players in safer zone closer to the 10-20% mark while minimizing sacrifice of minutes you'd like them to be on the court.

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