# 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. :)