Seven straight 3s, by the numbers

Last night I asked how common it was for a player like Ray Allen to make seven straight 3-pointers. Was he hot? Or is that just the kind of run good shooters have once in a while?

Research has never found the hot hand to be anything like as common as we assume it to be, and some studies leave open the idea that there is no such thing as a hot hand at all.

Several smart people have weighed in to help us assess Ray Allen's series-changing shooting in Game 2. For instance, Neil Paine of Baskeball-Reference has written a blog post that delves into numbers, then concludes: "Allen's streak neither proves nor disproves the hot hand ... It was simply a great performance by one of the most skilled 3-point marksmen in the game's history."

Meanwhile, Sandy Weil was a co-author of the preeminent research into the hot hand, and he e-mails to explain that depending on how you look at it, Allen had about a one-in-four chance of having a game like that.

He explains:

If you just ask: what is the chance that a 40% shooter makes 7-of-7? Then the answer is 0.0016 (about 1-in-610).

But I think that that is not the right question. I counter with this question: Wouldn't you be just as excited if he made seven in a row at any point during the game -- not just at the beginning?

If so, ought we not ask instead: What is the chance that a guy who takes 11 shots will make (at least) seven in a row in there somewhere? Well, then you give him the chance to start his streak with his first, second, third, fourth or fifth shot. So, you have to multiply the probability above by 5, giving 0.0082 (about 1-in-122).

But he missed some 2-point FGAs in there, so he didn't really hit seven in a row, did he? What if we talk about his making any eight of his 11 3-point attempts? Well, now there are 165 ways that we could sprinkle those three misses among the eight makes: 0.0234 (1-in-42)

But then we are still making another mistake: Wouldn't you be just as excited (actually more so) if Allen had made nine or ten or all of those 11 shot? So, we should probably ask: what is the chance that he makes at least eight of his 11 attempts? 0.029 (1-in-34)

But we should ask: what other information are we ignoring that is too inconvenient for the story? Too often, when focusing on the good, we ignore the offsetting bad. This makes the impressive performance look more otherworldly than it is. By focusing on the seven in a row, we had lost track of the three shots he missed and the other one as well.

During his "streak", if you consider two-pointers (which are conveniently ignored since they don't support the hot hand theory in this case), his game shots, in order are: missed 2, made 2, made 3, made 3, made 2, missed 2, missed 2, made 3, missed 2, made 3, made 3, made 3, made 3, missed 3, made 3, missed 3, made 2, missed 3, missed 2, missed 2.

So, this means that he never actually made more than four shots in a row at any time. (Though he did that twice.) And he shot only 11-of-20.

What do those figures look like?

If we use his 40% shooting rate, he has a 44% chance of making at least four in a row. And a 13% chance of shooting at least 11-of-20.

Another curious fact to consider is that he broke his own Finals record (shared). Ray Allen had played in eight career NBA Finals games, all against the Lakers. He made seven and eight 3s respectively in two of those games. This suggests that he may be a good matchup for the Celtics against the Lakers. I'd have to watch the film, but what if the Celtics, with time to game plan and prepare, run even more screens for Allen? Or what if Kobe is just a bit more likely to play the passing lanes than other defenders. It is quite possible that Allen and the Celtics do a good job of exploiting these kinds of things. Allen might just shoot a bit better against the Lakers than his career regular season 3-poing field goal shooting average of 40%.

Suppose that we look at him as a 45% shooter (his career all-shots success rate)? What kind of effect does that have on these figures?

Making seven in a row: 0.00374 (1-in-268)

Seven-in-a-row-amongst-11: 0.0187 (1-in-54)

Exactly eight-of-11: 0.046 (1-in-22)

At least eight-of-11: 0.061 (1-in-16)

At least four in a row amongst 20? 0.70

Shooting at least 11 of 20? 0.25

In this light, doesn't it seem a bit less otherworldly?