Duke vs. Butler: The Math Behind the Zoubek Free Throw Decision

Chuck AllenContributor IApril 12, 2010

Jonathan Daniel/Getty Images

If Gordon Hayward had hit his half-court shot at the buzzer, two things are certain. Indianapolis would still be partying, and Mike Krzyzewski would still be explaining his decision to have Brian Zoubek miss his second free throw with 3.6 seconds to go.

But was it the correct decision?

Allow me to set the stage. With 3.6 seconds left in the national championship game and the clock stopped, Duke’s center, Brian Zoubek, hit a clean free throw to give Duke a 61-59 lead. Zoubek had a second free throw coming. Butler had no timeouts. Duke was tired, its five starters having played most of the game. Both of Duke’s big men—Zoubek and Thomas—had four fouls. A partisan crowd of 71,000 was buoying Butler.

At that moment in time, Duke was likely to win.  But Coach K could not guarantee a win, no matter what he did. All he could do was try to improve Duke’s chances. That’s his job.

Given the circumstances, Mike Krzyzewski signaled for Zoubek to miss the second free throw.

So, let’s examine this mathematically. What chance did Butler have under each scenario? Was one scenario better for Butler than another? In assigning mathematical values, I have to use estimates. You may dispute these estimates, but I think most of you will find them reasonable.

SCENARIO 1 -- ZOUBEK MAKES THE FREE THROW

If Zoubek gives Duke a 62-59 lead, Butler has to do four things to win the game.

1. Butler must inbound the ball successfully. Absent tenacious defense, which Duke could not risk, teams successfully inbound the ball about 95% of the time. Since Butler needed a longer inbound pass toward mid-court, its odds of inbounding were probably about 85%.

2. Butler must advance the ball another 30 feet or so to the vicinity of the three-point line. Having only three seconds to do so, Butler had about an 80% chance of getting there. (Hayward did, in fact, advance the ball 30 feet in the last 3 seconds.)

3. Butler must hit the three. Hahn and Mack shoot about 40% from behind the arc, and Duke could not defend the shot closely for fear of fouling.

4. Butler must win in overtime. Given the massive crowd support, Butler’s fresher legs, and Duke’s serious front-line foul trouble, even Coach K thought Butler might win in overtime. I give Butler a 60% chance in the extra period.

Chance of Butler winning after a made free throw: .85 x .80 x .40 x .60 =  0.1632 (about 16%)

SCENARIO 2 -- ZOUBEK MISSES THE FREE THROW

Following a missed free throw, Butler trails 61-59 and must do three things to win the game.

1. Butler must rebound the missed free throw. Generally, defensive teams rebound 90% of missed foul shots. With Duke pulling one player off the lane, Butler’s odds were probably closer to 95%. There was no competition for the rebound. Duke was setting its defense.

2. Butler must advance the ball about 30 feet in 3 seconds, this time to mid-court. Again, I give Butler about an 80% chance of advancing the ball 30 feet in that time (same as in the first scenario).

3. Butler must hit a mid-court shot on the run.  Now, most of you have taken mid-court shots in an empty gym, usually from a standing or “set” position. You know how it goes. Out of 20 shots, 18 hit the rim and 1 or 2 go in. It’s easy to hit iron from 45 feet. It’s hard to score. Statistically, mid-court shots taken on the fly go down about 5% of the time. (Set shots are easier.)

Chance of Butler winning after a missed free throw: .95 x .80 x .05 = 0.038 (about 4%)

CONCLUSION

Based on these numbers, Coach K’s decision to have Zoubek miss the second free throw produced a four-fold reduction of Butler’s chance of winning … from 16% to just 4%.  Even if you believe that 10% of flying half-court shots score, Butler’s chances of success were still cut in half.

You have to ignore the fact that Hayward’s 45-footer “almost went in.” Most half-court shots "almost go in." And even if it HAD gone in, the decision to miss the free throw was still the correct one, statistically, without the benefit of hindsight.

ESPN recently featured a video about the science behind Hayward's half-court shot. The conclusion was that the ball would have gone in if it had left Hayward's fingertips further to left by an angle equal to the width of the aglet on a shoelace. ESPN wants you to believe it was desperately close.

I think the video makes the opposite point. To me, the video shows that it is SO hard to hit a half-court shot that, if you are off by even the width of an aglet, you'll miss.

Coach K’s decision was certainly not the conservative one. A three point advantage would have given Duke the comfort of overtime protection. And Duke had to avoid fouling at ALL costs. But, if you’re Duke and you’re looking to maximize your chances of winning, Zoubek needs to miss that second try.

CBS’s lead analyst Clark Kellogg knew this too. Unaware of Coach K’s decision, he made a telling comment. As Zoubek took the line for his second free throw, CBS’s Jim Nantz said, "Butler has no time outs.” Kellogg felt compelled to reply. "So they would hope that Zoubek makes this,” he observed. He was right.

Clark Kellogg and Coach K were both right. They both knew, intuitively, what the numbers tell us here.

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