Posted by Neil Paine on June 7, 2010
In case you missed last night's game, Ray Allen went crazy from beyond the arc in the first half, knocking down his first seven 3-point shots of the contest. As Henry Abbott notes, Allen is a career 40% shooter from beyond the arc, on 6678 career attempts. So Henry asks:
"If you hit 40% of the time, and take 6,678 shots, how often would you end up with seven or more makes in a row?
Does Allen do that more often than you'd expect? (Bring on your probabilities!) If the answer is yes, then let's talk about the hot hand. But if the answer is no, well then let's appreciate this is the kind of night good shooters have sometimes, even without the supernatural."
Let's address the first question -- on average, how many runs of 7 straight 3-pointers would you expect a Ray Allen-like career (40% shooter, 6678 shots) to contain? And what's the probability that he would do it at least once in his career? Well, there's no easy way to set this up in equation form, because you have to account for the possibility of multiple sequences containing at least 7 consecutive makes, which would require some heavy-duty combinatorics. Instead, when confronted with a problem like this, I like to set up a Monte Carlo simulation and derive the probabilities by running a large number of trials (for other examples of posts where I did this, see here, here, here, here, and... well, you get the idea).
With the experiment set up to simulate five thousand 6,678-shot "careers" with a 40% chance of making each shot, we find that (on average) each career contains this many streaks that ended at each length:
So if you hit 40% of the time and take 6,678 shots, you can expect to have about 6-7 stretches (6.5, technically) in your entire career where you have seven or more makes in a row. Also, you'll have at least one stretch like that in your career 99.9137% of the time -- in essence, this means a "true" 40% shooter is virtually guaranteed to have at least one run like Allen's in his career due to chance alone. These stretches can come in one half, one game, or even across multiple games; in fact, we find that the best streaks of all time (Brent Price & Terry Mills in 1996, not coincidentally when the arc was shorter) made their 13 straight across several days. But how many times has Allen rattled off at least 7 in a row during his career?
Unfortunately, it's impossible to answer that question with our current dataset; you would need a play-by-play log of every Allen game to determine exactly how many times Allen has had a streak of each length. But if anything, it seems like it happens less than you'd expect due to chance -- if you're a true 40% shooter with a career as long as Ray Allen's, a stretch like last night's should happen roughly every other season. But in real life the odds of making your 7th in a string of 7 straight are probably less than your odds of making the 1st, simply because the defense will have made some adjustment to stop you from doing what you're doing. This would mean they were under the belief the Hot Hand exists (otherwise, they'd be just as content to take their chances with 40% on the 7th attempt as they were on the 1st).
Also working to drag down the chances of making 7 straight in real life is the tendency for players who make multiple shots in a row to force subsequent shots in the belief they were hot, only to miss those "heat-check" attempts at a higher rate than normal. In other words, Ray Allen the 'true" 40% shooter might think he was on a run that boosted his true ability to, say, 50%, so he takes a shot he normally makes only 30% of the time on the next possession... but quickly realizes that it's still a 30% proposition. Only when Ray stays within his normal shot selection is he a "true" 40% shooter, and that shot selection can be changed by his own choices, as well as those of the defense.
Either way, though, we can say with some certainty that a streak like Allen had last night is indeed possible due to random chance alone. While it's unlikely that any given stretch of seven consecutive 3-point attempts would produce 7 makes (the probability is 0.4^7, or 0.1638%), if given enough stretches of seven consecutive shots, you're inevitably going to see some runs where he does in fact make all 7. That it happened in an NBA Finals game -- and not a meaningless Celtics-Nets game in February -- was all the more beneficial for Boston, but 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.