The (Opportunity) Cost of Shooting Free Throws

Mark Zerof-USA TODAY Sports

Getting to the line and shooting some free throws is always a good thing...right?

I think free throws are interesting.  To be more specific, I think possessions that result in free throws instead of a field goal attempt are interesting.  The reason is the tradeoff that occurs when attempting free throws rather than a field goal.  Depending on how you view that tradeoff, you can arrive at two completely opposite conclusions about which result is more preferable.

Just to be clear, I'm talking about the times during a game when a Wildcat goes to the line for a couple of free throws instead of getting a shot attempt.  Maybe they got fouled in the act of shooting or it's late in the half and the opposing team has put the Cats in the double bonus.  For simplicity I'm going to leave out one-and-one situations and three point shots - this is all about 2 free throws versus 1 two point attempt.

The opportunity cost for attempting free throws is that you don't get to attempt a shot.  To score two points from the line you have to get the ball to go through the hoop two times.  To score two points on a shot you only have to put it through once, and you have the option of trying to accomplish that standing much closer to the rim. On the other hand, you can shoot free throws without a hand in your face.

A trip to the line is usually considered to be a universally good thing, but I always wonder if the team would have been better off if they could have gotten a shot instead.  That wonder tends to increase when a free throw gets missed.  To put it another way, if you had the option of declining free throws and getting the ball with a new shot clock instead, would it be advantageous to do so or is it always better to go to the line (aside from obvious situations like being down 3 with only a few seconds remaining)?

I always wonder if the team would have been better off if they could have gotten a shot instead.


Let's say Willie Cauley-Stein has the ball on the block and is about to turn around and dunk it, only to be fouled by an opposing forward (I'll call him Chane) and sent to the line for two shots.  Now, WCS is no Jodie Meeks so we would all much rather see him dunk the ball instead of shooting free throws.  Most commentators would declare this to be a "smart foul."  There is some value in tagging Chane with another personal, but we're not going to worry about that here - this is all about putting points on the board.


FT% 2pt FG% At the Rim FG%
Willie Cauley-Stein 45.7 65.6 81.4

You can see Willie's current shooting percentages above, with the last column listing his FG% on shots that come around the basket (dunks, layups, tip-ins) which is available on Hoop-math.  The natural question to ask is: how much does the team lose out by WCS going to the free throw line rather than take a shot? In other words, what is the tradeoff between two free throws versus one field goal when #15 is the player involved?

There are essentially two ways to answer that question.  The first is to consider how many points Willie would score on average in each situation.  At 45.7%, with two free throws we expect him to net 0.457 * 2 = 0.914 points whereas a field goal nets 1.312 points (.656 * 2).  So, on average, Willie's trip to the line costs the team 0.398 points each time it happens.

Now obviously you can't actually score 0.914 points on a couple of free throws or 1.312 points on a field goal.  Those are averages over the course of many trips to the line and many shots attempted, but they can't be attained on any single possession.  So the other way to look at this tradeoff is to examine the probability of Willie scoring different amounts of points in each situation.

Probability WCS scores: Free Throws 2pt FGA At the Rim
0 points 29.5 34.4 18.6
1 point 49.6 - -
2 points 20.9 65.6 81.4

A field goal is all or nothing - you either get two points or you don't.  But free throws give you a chance to mess up once and still come away with a score.  For as bad a shooter as WCS is, there is still a 7-in-10 chance that he comes away from the line with at least one point whereas there is only a 6.5-in-10 chance of doing the same on a two point shot.  If he's around the basket that increases to an 8-in-10 chance.

If the goal is to come away with any points at all he's actually slightly better off getting the free throws, unless the shot is around the rim.  HOWEVER, if the goal is to maximize the points scored he's better off taking a shot than going to the line: WCS going 2-2 from the free throw line is about as likely as him missing a dunk or a layup.  He would need to shoot 81% from the line to have the same chance at scoring two points from the line as he does from the field, 90% for the same chance as a shot at the rim.

This is what I find so fascinating about free throws vs field goals. Willie Cauley-Stein is as extreme a case as you will find - a player who is amazing at making shots but is very poor from the line.  Yet even in his case you can make an argument that the free throws are the better option.  It's not a strong argument to be sure, but there are situations when the team might be better off (from a scoring perspective) with him getting to the line even with his poor shooting.

The situation becomes less clear cut when we look at a better shooter.  Here's Julius Randle:

Probability Randle scores: Free Throws 2pt FGA At the Rim
0 points 8.4 43.8 26.2
1 point 41.2 - -
2 points 50.4 56.2 73.8

Julius is roughly 5 times more likely to come away empty handed when attempting a shot than when he goes to the line, 3 times more if he's around the basket which makes free throws the better choice.  On the other hand, he has a slightly better chance of scoring two points from the field than he does from the line and a much better chance if he can get a layup or dunk which gives the edge to the field goal.

So which scoring opportunity is better?  The answer depends on how much you are willing to trade a chance for multiple points for a chance to avoid not scoring anything at all.

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