To Foul or Not To Foul: An Analysis
To Foul or Not To Foul: An Analysis
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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 twopoint 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.
 Nontraditional (NT): Foul Kentucky (hopefully a bad freethrow 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 nontraditional 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 NonTraditional (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 xaxis and the Kentucky scoring chance along the yaxis. 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 nontraditional 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 lastsecond shot (no surprise).
Given the efficiency of our offense, and the relative nongoodness 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. :)