To Foul or Not To Foul: An Analysis

Submitted by Swayze Howell Sheen on March 31st, 2014 at 4:34 PM

Reader will brought up an interesting question in a board posting recently: should Michigan have fouled Kentucky with about 20 seconds left, putting them at the line, but (critically) giving Michigan the ball back with a chance to tie or win?

To my surprise (especially given this crowd), there were a lot of "gut" responses based on feelings, emotions, and in some cases, how such options would be hard to explain in the media. 

So I did a few small calculations. The simplifying assumptions were these:

  • Kentucky has some chance of making each free throw (call this Kft)
  • Kentucky has some chance of scoring when we don't foul them (Ks)
  • Michigan has some chance of scoring if they have the ball back (Ms)
  • There are only two-point baskets (no threes for simplicity)
  • If the game went to overtime, odds are 50/50
  • On a missed free throw by Kentucky, Michigan gets the ball 100% of the time
    (clearly a stretch in this game)
  • If we let Kentucky play it out, they will get one chance to score and the game will end either with them winning or go to overtime.
  • If Michigan gets the ball back with plenty of time, assume they either score (as dictacted by Ms above) or miss; no free throws, etc.

With these assumptions in place, we can start to calculate: what should Michigan have done to improve their chances of winning the game?

There are two options we will compare:

  • Traditional (T): This is what we did. Play defense, and hope Kentucky misses.
  • Non-traditional (NT): Foul Kentucky (hopefully a bad free-throw shooter) and get the ball back with a chance to tie (if down two), or win (if down one or still tied).

Consider the traditional approach first. Let's assume that Kentucky has a 40% of scoring to win the game in the fashion they did. Thus, 40% of the time, Michigan loses in regulation, and 60% of the time, it goes to overtime. By assumptions above, Michigan's win probability in this case is 30% (half of the overtime outcomes).

Consider the non-traditional approach, which is trickier. Assume here a low rate for Kentucky free throws: 50%. Thus, 50% of the time, Kentucky will miss the first free throw, and Michigan gets the ball back with a chance to score and win; assume again a similar 40% chance Michigan scores when they have the ball. Correspondingly, 60% of the time, the game goes to overtime with 50/50 odds. Thus, on the first miss, Michigan has a 70% win chance. 

Unfortunately, 50% of the time, the Kentucky player makes the first free throw. There are two further cases to consider then. If they miss the second (which happens 50% of the time), Michigan has a 40% chance of winning in regulation, but 60% losing. If they make the second, Michigan just has a 40% chance of sending it to OT, where they have a 50/50 shot.

If you add all of those win probabiities up, the Non-Traditional (NT) approach, assuming the numbers above, has a win probability of 50%, which is 20% higher than the traditional approach (T). Thus, assuming the numbers and other things above, fouling was the better option.

However, that is a pretty low free throw percentage, and the chances I gave of Kentucky or Michigan scoring a basket (40%) were chosen arbitrarily. Thus, I varied each of these and produced the following graphs.

This first graph assumes the 50% (Kfs) as above but varies the Michigan scoring chance along the x-axis and the Kentucky scoring chance along the y-axis. Results in BLUE mean that Michigan would have increased its chances of winning with the NT approach; RED means a decrease by fouling early. The value shown is the difference in win probability between the two approaches.

As you can see, the (x=40,y=40) point shows the 20% increase calculated above.

I also made a graph assuming that Kentucky shoots free throws at a 75% rate, not 50%. It looks like this:

As you can see, it looks a bit different, with the non-traditional approach (foul early and get the ball back) not doing as well.

More broadly, what you can see from the graphs are this: if free throw shooting is bad, fouling early makes sense, especially if you have a good offense with a good chance of scoring. Fouling early also makes increasing sense if the other team is likely to make their last-second shot (no surprise). 

Given the efficiency of our offense, and the relative non-goodness of Kentucky free throw shooting, I think we did the wrong thing.

Of course, I reserve the right to be wrong in the analysis (it was a little hastily thrown together); critque away, as you always do. :)




March 31st, 2014 at 4:51 PM ^

I think the problem here is the assumption that any missed free throw is rebounded by Michigan.  Although the non-shooting team has automatic advantages when it comes to rebounding, given the fact that Michigan rebounded fewer than half of Kentucky's misses, I can't see this being any higher than 75%, and I think this makes the traditional approach correct.  The worst possible scenario is one made free throw, a miss, and an offensive rebound, and that was a sadly realistic scenario.

Thanks for putting this together though.  :-)  I applaud the measurement of non-traditional approaches.

Go Blue!


March 31st, 2014 at 4:55 PM ^

given the teams we watched yesterday, I think the biggest flaw in the assumptions is the 100% that Michigan gets the ball back after a missed free throw.  very interesting analysis though.


edit to add: beaten to the punch with my comments!


March 31st, 2014 at 4:55 PM ^

was dread knowing that Kentucky had the chance for the last possession to win the game, where Michigan could only defend and hope for OT. While I think the shot they gave up was the best-case scenario with less than 10 seconds remaining, I agree with your analysis that Michigan was probably better off giving 2 quick fouls, giving Michigan the ball and a (likely) deficit with 20 seconds remaining.

To follow up on that, I think the made 2 on the other end with 30 seconds left was a negative in disguise. After Stauskas misses the 3, Michigan should have probably held the ball to ensure the last shot to tie or win. This negates even having to foul to get the last shot. Then Michigan dribbles it down to 15 seconds, calls timeout like we've done this all season, and runs a play for the tie or win.

Is that a better chance of at least forcing OT than the three Michigan conceded anyway? Who knows, but at least then it's Michigan's crazy-efficient offense that decides the game, not their mediocre defense.

snarling wolverine

March 31st, 2014 at 5:55 PM ^

Gotta disagree.  When you're down, you want to extend the game as much as you can.  Shooting with 30 seconds left leaves you those 30 seconds to figure something out - whether it's to foul, go for the steal, whatever.  If you dribble it down and take a last-second shot, that's it - if you miss, you go home.  

If you're tied, you can dribble down, but not when you're behind IMO.  Likewise, I suspect that if Stauskas or Walton had made the 3-pointer on our last real possession, UK would have quickly tried to score.  (I'm still surprised Walton missed that one - he'd been shooting so well in the tourney and that was a great, open look.)



March 31st, 2014 at 5:01 PM ^

Good analysis, thanks.

One thing absent from the analysis is an assessment of M's ability to force them into a difficult shot, which they did.  If they were confident they could do that the traditional approach was the right call.  The guy took a shot he had a much less than 50% chance of making and he made it.  That's a low probability outcome but one we can live with.


March 31st, 2014 at 7:53 PM ^

The innate ability to do so wouldn't be different, but the circumstances - late game situation, freshman ballhandlers, maybe there's something you can do to rush them a little under those conditions to force a lower than average percentage shot.  If you think so maybe that could swing you toward not fouling.


March 31st, 2014 at 6:41 PM ^

I think we should have fouled...we're not a real good defensive them the last shot is a riskless situation for them (make their freshmen have to make hard decisions).

Given our offense, I'd rather have the last possession.  We put ourselves in a no-win position.


March 31st, 2014 at 7:31 PM ^

I wanted us to foul early. Even if UK got a rebound off the missed FT we would have fouled again immediately Still leaving more time on the clock. 

They got the ball back, game tied with 20+ seconds left. They were scoring at will, inside, all game. I was shocked they did not attack the basket with about 8 seconds left (they seem to be getting ALL of their rebounds off the rim) and no offensive foul calls. 

They were shooting 60% (6 out of 10) from 3 before that final shot. So even allowing them to shoot from outside was risky. 

No matter how they would score they would make sure we had little time to get up court and take a good shot to tie or win. 

Fouling, in my opinion, greatly increased our chances  Our defense is our weakest link yet we bet it all on making a defensive stop.

Mercury Haze

March 31st, 2014 at 8:44 PM ^

It's not unprecedented.  Jim Valvano was a believer in fouling while tied, including in the 1983 championship game.  He wanted his team in control at the end.  


March 31st, 2014 at 9:21 PM ^

based on the situation, I don't think fouling at the end was the right call.

A) The way Kentucky played it, they kept the ball on the perimeter in the hands of a player who was a much better FT shooter than any of the bigs.

B) Odds of making a last-second shot when your team is down seem a lot less favorable than when the game is tied (opionin based on nothing--would like to see stats on that though).

C) Unless we did something crazy like sub out all of our ball handlers and put in every single big we had, odds of rebounding a game-deciding rebound off a missed FT seem low.

However, I wish we had fouled them down low more often at earlier parts in the game. I feel like we would have limited to fewer points by fouling their terrible FT shooting bigs and potentially rebounding off a missed FT than what we could be playing them straight up.

An Angelo's Addict

March 31st, 2014 at 9:22 PM ^

I was actually really thinking they should have fouled again following the inbounds pass before the final 3 point shot. Were they already at the bonus? Because I thought we still had 1 more foul to give and with another foul, it would have made them make a play with just 4-5 seconds left


March 31st, 2014 at 10:05 PM ^

who was on the floor but the Harrisons are .80+ FT shooters. There was a much better chance that they would miss a long, contested 3 than a FT. Had Kentucky tried to get the ball inside maybe they could have fouled a less proficient FT shooter while jockeying for position but they didn't do that. It seemed like the play was to spread the floor and throw it up there with enough time for their bigs to fly in for a put back.

Caris' first foul was earlier than what JB wanted. He wanted the foul after 10 seconds not at 10 seconds as Caris did. That gave Kentucky too much time to hunt a comfortable shot. He also wanted to keep Kentucky out of the paint which they did...although I don't think they pressed that issue.

When all is said and done one can't forget that Michigan actually did get some stops on their way to tying the game. I don't think it was an impossible proposition that they get the final stop. I'll take a contested 3 from a guy shooting them at less than .40 than a FT where his average doubles his chances.

steve sharik

April 1st, 2014 at 2:33 AM ^

"The plan was for Aaron to shoot a three b/c they won't guard you (paraphrasing)" is a bunch of BS.  I think their plan was to simply shoot a jumper with ~5 sec. left and get a put-back.

As for this discussion, I think we should've zoned up and left one of their bigs open initially, then fouled (assuming fouling was the plan).  A buddy of mine was suggesting we start playing hack-a-shaq with about 5 minutes left.


April 1st, 2014 at 10:45 AM ^

Interesting discussion here. When I started reading, I was hoping the OP would use the Points per Possession/offensive efficiency stats for each team for this game to help guide/determine which option would be better. In other words, based on 19:40 of this game, what is the true probability of UK scoring on that last possession? What had their FT percentage been to that point? While the above is an interesting way to look at it, I feel like you could use even more accurate data to get an even finer picture.

That being said, it's tough to second-guess this one, in my opinion. You've got a 35% 3-pt shooter (who had been shooting well in the second half, however) who takes a long (nba range), contested shot. That 35% is probably mostly on open looks, on the line, etc. You've got to like the raw chances of that shot missing, right? Rebounding that miss would be another story, and I agree with above comments that the put-back was the true play here.


April 1st, 2014 at 12:33 PM ^

In hindsight it seems like fouling would've been good.  We would have had the ball with 20ish seconds left.

I think on that last possession we really did everything we could.  LeVert had his hand in the guy's face and sometimes you just make those shots.


April 1st, 2014 at 11:32 PM ^

This team ended where it was supposed to. The only "except" I'll put in this small endorsement of your post is it could have applied to the entire season. Were we supposed to win the BIG by three considering(and all have considering contained therein).  No to ? #1.   This should be tossed in just to make things a bit less murky:   GONE:  POY, Hardaway Jr., and by every announcer that called a game in last year's tourney, "the second coming of Bill Walton."                            ^In essence, given the above, we should not have won conference, especially by 3, we should not have been able to prove our final poll no: 7, where we basically ended, and to get this far and-let's be honest- a realistic, perhaps too obvious chance of once again playing in the final couple- had to toss that in there for fans such as I was some 25 years ago.where I could not take a loss of this magnitude w/o a 7 day Grace period of not speaking, not just about the game, but speaking period in my normal manner.                                  ^We all know what vision is in hindsight, so further not be said there.  All that really needs to be stated, imo, is this man will take whatever team he is given as far as it can possibly go, as he did this season.  Could they have won? Certainly. But that can be said as to TN and many teams we played last year.  Things end when they should.   The only thing we need to embrace, much like we did when we had Bo running the ship in fb, minus obvious "homerism, our coach will take us as far as the talent on hand.  Could Bo have won some of those bowl games with a little more creative offense? Certainly. But could any other coach in the country have taken the average attendance from 67K to stand in line at over 103,000 weekly? No, he was the perfect fit and all success thereafter can be attributed to him.    Coach B is freakishly close to that status, probably moreso because BB features but a handful at a time and we are all able to see how he turns a possible  3.5 star into a 4.5 at tourney time. Bo did the same with all the "3 stars" he recruited, but playing as 11 made them all appear to be h.s. AAs.  We're exacty where I want to be. I loved it when we had the best fb coach in the country with no court appearances to disrupt the cohesiveness so hard to come by, and I love it now that Beilien is able to do the same in Cazzie's home- not sure if he still has the deed- but I'm sure I love where M BB is and where it's going. He's as close as I've seen to the Real McCoy as Walter Brennan. Don't second guess my friends. Just think, "Damn, yet another marvelous job by that overachiever."  He is all of that and we're damn lucky to be able to say he's ours.

Indiana Blue

April 2nd, 2014 at 11:04 AM ^

The first kickout to Nik inthe corner was the best opportunity.  Nik had 2 guys flying at him.  If he shot fakes, then moves to a position where the defensive man is going to hit him and then lets fly .... we have Stauskas on the FT line with 3 FT's to take the lead.  Oh well ....

Go Blue!