More videos with Professor Goldberg: bit.ly/LisaGoldberg

They should get a thermal camera and record the players' hand temperatures while they play.

My problem with this is if you did permutations on someone making 9/12 shots you could argue their sequence wasn't 8 or 9 in a row so therefore not on the high end of the "hot hand spectrum". But they did have a hot hand because they were shooting a higher percentage that game than normal. I think the study is incorrectly defining what we mean by hot hand.

I think how you define "hot hands" is really important in a study like this.

It's great that the Professor emphasized that you bring your own rules to an experiment and so you get an answer based only on these rules. Changing the rules may change the result so it all comes down to the compromise of what initial conditions we choose.

i love it when people dispel the false notion of hot handedness, that being said, shouldn't they be comparing "hot" games the a random sequence given their season FG%, not the game's FG%? If a player makes 25/26 shots in a game, the only sequence that would be considered "hot is if he missed the first shot, then hit every shot afterward, when a colloquial definition of "hotness" would say the player was obviously hot because they hit every shot

One thing that they fail to mention is that when someone has a "hot hand" they'll often times execute a "heat check", taking a far more difficult attempt than they normally would. This would surely skew the data.

I think this study faces yet another flaw. I'll use the game at 16:05 as an example. You are comparing the game with all the possible ways to score 31 out of 44 throws. But had Klay Thompson not have a hot hand, he wouldn't have scored 31 (maybe 25 or 28). So by doing the study this way, you are considering his point average as normal. I would suggest comparing his usual chance of scoring in all games with his chance of scoring during his streak.

Another flaw in this study, aside from some of the others stated, is that it focuses on the top team and the top shooters in the league that already hit a disproportionate number of shots. If JR Smith all of a sudden hit 11/14 threes in a game, you can't deny he had a hot hand that night. This study completely eliminated phychology from the equation and treats a shot as a random independent occurrence devoid of a human element.

What about just hotness as increased shooting percentage and not streaks? Was his shooting percentage for the game a random fluctuation or was he really shooting better? Klay Thompson's field goal percentage that season is about 47%. Using a binomial distribution with p=.47, we find the probability of clay shooting that well to be 0.0005, or 1/2000. He has not played that many games. One should look at all games and see if the distribution matches what would be expected from a binomial distribution.

i think the meaning of being hot is a high FG%, instead of the actual streak of made shots. For example, if a player makes 21/30, we'd say he was scorching hot that night, even if the made shots werent one after the other.

The scrambling test literally made me drop my jaw in awe over how genius that concept is.

I really like Professor Goldberg--it's good to see her again.

Wouldn't it be easier to see hot hands in a bad player than a good player?

The problem is that streaks of hot and cold shooting are not broken but one miss or hit.

Not sure if the rendering/KZfaq has had some part in it - but this video seems quite washed out - no contrast!

This phenomenon was touched on in Gilovich's excellent "How we know what isn't so". A categorically recommended read for anyone who wants to understand how and why our brains make mistakes.

I wonder if the odds you take the shot go up while the odds you make the shot stays the same.

Why are you treating a basketball shot as if it were a random event and not an application of the player's skill?

That guy near the end had 3 streaks of 4 in a row and 1 streak of 6 in a row. I don’t watch basketball, but that sounds like a hot hand to me. I think there’s some misinterpretation in what a hot hand is

15:55 I died a little inside when she said they didn't distinguish between foul shots, 2 pointers, and 3 pointers; different probabilities and changing proportionality. Nevertheless, one player scoring ~60+ points a game is a less effective way of winning; teams occasionally funnel the ball to a distinguished player on a seemingly good night out of player respect, fanfare, and adrenaline.

Brady, you're really on a streak with these great videos!

I understand the Gambler's Fallacy because it is mostly about luck when gambling, but basketball is a game of skill. By just looking at the data you are ignoring the skill and psychological aspects involved.

I think this is not an arena for solely interpreting events using math. He might miss a shot but it is because a guy who has a hot hand will be guarded more fiercely too. This creates more space and more likeliness of finding an easy assist. (The hot hand is not solely about making buckets. It is about creating space.) John Starks in the 94 Finals is the perfect counter example of the hot hand- the cold hand. His poor shooting basically suffocated the lane. And, he continued to choke for the remainder of the finals.

basketball is just coin flipping - you either make it o r you don't. There is no skill or strategy involved. That's why it's perfect for statistical studies.

One of my favourite Numberphile videos. That being said, I think in basketball looking for a hot hand shouldn't just be observed within a game, but it might also be very informative if the same checks for single players in each game were done for many players, but across a whole season. I think by looking for streakiness not just compared to the rest of the same game (which probably has a Hot hand being carried from earlier streaks), but should be compared to an average game for that player in some way. Or maybe across a season we could compare the number of full games where a player is hot, using a permuted season divided into random games as the Null Hypothesis / baseline. As a basketball fan I feel llike this bias would yield a much different story.

Seeing NBA Jam appear in a Numberphile video has to be the greatest moment of the year.

This video is soooo good. Thanks numberphile!

If I understood right, a player hitting all of his shots in a game won't have a "hot hand" by this calculation. that is obviously a bad definition of "hot hand". Same if the player only misses one shot. Missing the first shot would be a hot hand and missing one in the middle of the game would not, by this calculation. But obviously the player is hot regardless. C'mon if scoring 60pts is not being hot then what is?

typical viewer: “They have all this data and methodology, but my FEELINGS though.”

So, if you take 20 shots and hit all shots, then after random scrambling there should be 0 permutation from the average. Aka, not a hot hand. Should you not compare this string of 1s or makes against "the average" instead of against itself?

HHT and HHH where counted as one measurement, even though they have 2 outcomes each. If you count them according to the number of outcomes, 2 each, then it is 4/8 = 50%

TRIAL DESIGN ERROR!!! What would happen if someone makes all his shots and goes 20/20? Null hypothesis and a "streaker" would be the same, wouldn't it? By all human definitions, going 20 for 20 is REALLY hot! 19/20? 18/20? 17/20?

I don't know why but I love the term Null Hypothesis. It's music to my ears for whatever reason

I'm an engineer... I appreciate science, and I don't usually use ''feelings'' as an indicator of anything ! But ... When you play ball, sometimes you just feel it !! The rim just seems bigger !! Something in your mind when you kind of achieve a level of focus, the game slows down !!! Yea I know I'm not disproving anything here but anyone who played ball knows the feeling !! Numbers don't lie, but they often don't tell the whole story either !!! Great video ! I wish numberphiles would do more '' sports stuff'' !!! Cheers !

This was a great study that I appreciated reading when it came out overall and love seeing her on the channel! My only "issue" is that the study doesn't seem to take time and frequency of shot into effect which are large parts of the "hot hand" which is why I would guess many coaches and players have problems with these studies. You're not taking into account if these are shot in succession or with a small vs large number of possessions in between. So this data looks at 2 shots in 2 successive possessions to be the same as a buzzer beater halftime 3 being shot "next to" their next shot being taken a few minutes into the 3rd quarter and the same as 1 shot being made then the player not shooting for the next 6 possessions (for example) and then making another which due to time and number of possessions in between wouldn't be considered a hot hand generally. Also, no one would consider this 2nd option a hot hand but the study could consider it to be one, while the 1st option could be considered the beginning of a hot hand by any natural observer/logical argument.

very interesting. i’d love to see more studies on this, although last years Warriors are probably the best case study available. From a viewing perspective, that was the hottest shooting team in the history of the sport. The only other team that would be close would have been the 2015-2016 Warriors.

Another thing to consider is that it is easy to remember the successes than the failures, especially if a failure quickly follows a success, or something dramatic happens after the miss, such as the opponents moving the ball to the other side of the court. As such, that could skew people's perception of a scoring streak, making them appear longer than they are. It's like how you can read a book for what feels like 5 minutes, but ends up being several hours.

No basketball shot ever has 50/50 chance of going in.

I appreciated her explanation at the end that the hot hand can't necessarily be explained by the parameters of her study. I've watched Klay's 60 point game a few times, and while he made a few tough shots that would normally be attributed to having the hot hand but could be explained by randomness, what I found most striking about his performance that game was his uncanny ability to get open. It was like he was thinking the game two steps ahead of everyone else. I've never seen him - including the 37 point quarter! - play a game so effortlessly. His makes vs misses weren't what defined his hot hand that game for me, it was everything surrounding the shots.

10:20 So this effect is about grouping results together and combining the relative results instead of the absolute ones, which would be 4 heads and 4 tails with an average of 0.5. I guess for this to happen it is necessary that the grouping is not random (higher chance of more than one H in one group). Very similar to gerrymandaring.

This is a great video, you should make more videos were people explain their own experiments/papers and the stories behind them

one trouble I have with this is the mathematicians are assuming scoring a goal or basket in sports is a purely random event. It is possible that on that day in those conditions the variables have aligned to allow one particular player a better than average score rate.

I love that they keep using the 1967 finals footage as the video chapter title background. Great series

I find that videos on sports statistics tend to be hit or miss.

I love when math is given real world representation, makes it easier to understand for those of us (me) that were never too hot with math!

A bit extra from this interview: kzfaq.info/get/bejne/eNx3Zpil3bfPdGQ.html The original Hot Hand paper (by Tversky and others): bit.ly/2HJNFeT And the paper which debunked it: bit.ly/2HuIoKB And the newest paper about the Warriors' hot (?) hands: bit.ly/2Ha3nCR

I'd be interested in seeing a sequence of 1&0s that would be considered a hot hand

More of a psychological problem than a mathematical one. "In the zone" literally means having the most optimal mental state for the given kinesthetic activity. Whether that translates into "luck" or not, now that's the mathematical part of the problem.

Something that would go along with this is in the shooting sports. Lones Wigger a Smallbore shooter talked about the mental average. You would start out shooting really well but your mental average would cause you to start shooting poorly to bring your score back to your mental average.

I love how she doesn’t make any rash claims about her research and simply presents it as evidence. THIS is how everyone should do science.

Brady, I love your videos. As an avid basketball player and a mathematics graduate, I think these studies are not measuring what basketball players and coaches may be recognizing as having a hot hand. These measure temporal hotness. Something that comes and goes with any given shot. From experience, there is definitely such thing as having a hot hand. However it typically means that I am simply shooting with a much higher percentage than normal. If you are confined to a particular game, you are measuring within my “hot zone.” So I would expect general randomness. What I would like to see is a study of “hot games.” When a player has a hot or a cold game, is it just part of the general randomness you would expect from an X% shooter, or were they actually hot or cold that day? I suspect that outside factors such as sleep, injuries, diet, personal life... all play a roll on making a player extra hot or cold on any given day. When trying to implement a strategy around this, the trick is probably differentiating who is actually hot or cold vs who is simply experiencing a moment of randomness.

"Clay has made 40 points so far, but actually he's having a cold hand, let's stop giving him the ball"

I feel like we're missing something by not comparing the players scoring in each particular game to that player's average. If a player is suddenly making 80% of their shots when their overall average is closer to 50% of shots made, that's a huge difference. Additionally, the timing thing seems incredibly important to me. Are those streaks being broken after a break in the game? Are they being broken when the player hasn't had a chance to make a shot in a while for whatever reason? What are the conditions around the streaks being broken? At what point do the conditions to continue a streak no longer exist?

I would think that moving averages of some kind need to be involved. Missing a single shot in the middle of a streak shouldn't be so penalized when it comes to quantifying "hotness" (at least as I understand the conventional definition of it)

You seem to assume the chance to hit remains the same for each shot, but other things such as skill, morale, general well being also plays a huge role in the chance a player can make a shot, its pretty much impossible to calculate that.

In the coin flipping example, why is it interpreted as 6 measured groups of 3 coin flips rather than the actual 8 measured individual flips? When looked at as individual events, there are 4 of each for even odds as expected.

So timing ie fast actions could be interesting. I also think what happens post a missed shot is important. How bad was the miss? Rebound and score (assist) vs other team scoring.

This is a terrific video because it helps to explain what I’ve come to believe is the fallacy of the “hot hand” in sports. But more than that it alerts me to fact that small sample sizes may behave differently that big sample sizes, and that most problems are more complex that appreciated. Human aren’t likely good at understanding randomness, but we are good at picking out patterns. The problem is those patterns may in fact be misleading.

Coin flipping is different from a competitive endeavor like a sport. People playing a sport or other competition can find themselves "in the zone." It is a mental state. A confident state of mind. A self-hypnosis of telling your unconscious that we will do well today. There are no thoughts about other than the current game. No distractions. No relationship problems thought about. No money worries come to mind. No teammate problems. Feeling good. There are days when players are fully in the game; these are days when players are above their average. Days, one might say, that they had a hot hand. If a player normally makes 60% of their shots then making 40% is a not a hot-hand day. If a player normally makes 30% of their shots (they're on the team for defense) but today makes 40% -- bingo! a hot-hand that day. Depends on what is meant by "to have a hot hand." If a player is having a lucky streak -- not a skill streak -- shouldn't we avoid giving the ball to him? Skill is so much more dependable than luck. Say a player makes 6 shots out of 10 attempts. When scrambled (1001110011) there a fixed number of each length of streak. In the example one of each length 1, 2 and 3. If a player scores 7 out of 10 the average streak length must be longer. A "streak" -- a "hot hand" -- may well indicate higher than ordinary skill, and not just luck.

Absolutely excellent, brilliant formulation, very accessible.

Can the permutation test actually disprove the 'hot hand' though? If the additional 1's in a streak were caused by a hot hand, then the permutation sample will include those additional 1's, so should be biased towards having more ways in which that could have happened through random chance.

Wonderful video. Absolutely fascinating.

While many might be prone to believe a streak represents an underlying tendency more than it does, there is plenty of reason to believe that humans undergo periods of higher performance. We have objective good and bad days, we undergo constant neural plasticity, so I would say careful not to be too eager to flatter yourselves that peaks don't happen just because of short interval constancy.

I LOVE me some probability theory! Thanks for this! I would like to see a similar discussion in combat sports, like boxing or mma, that demonstrates how much is luck vs skill in winning belts.

I would have thought that statistical significance of a hot hand would be less related to single strings of shots, and more about periods of statistically higher percentages, so something more like a standard distribution and deviations based on a player's usual shot percentage over a period of time. Aka, a prolonged period more than a standard deviation above their usual shot percentage would be considered a hot streak, and a standard deviation below would be a cold streak. Also, it's probably worth considering that from the standpoint of this study having a hot hand is not the same as being in the zone, and perhaps that is why it specifically looks at the significance of shot streaks instead of percentages or other stats to determine whether it's there. Because the stats do seem to show that some players have significantly different shooting percentages between a good and bad game which is why they are thought of as streaky in the first place. What this study says is more that the idea of being on fire with your shooting doesn't necessarily mean you will hit long streaks of shots and mostly seem to show that when a player is in good condition they hit a higher overall percentage, but the statistical significance of the actual streaks doesn't change much. Being in the zone is essentially the idea of being in the perfect mental state while playing, while the hot hand refers specifically to score streaks. Similar ideas, but not the same, and showing that there isn't a statistically higher rate of shot streaks doesn't account for the overall higher level play that someone who scores 60 points in a game has, and streak or not 60 points is definitely a statistical anomaly. I suppose that's all part of what makes statistical analysis of sports interesting though, because humans and their mental states tend very much not to fit neatly into data sets.

I really think that you're looking at hot hands from the wrong angle. We need to collect information from people if a player was having a hot hand right after a game (without commentary). Then we can try to train an universal appropriator to predict "hot handedness".

Not a sports fan, so I don't know this 'hot hand' mythology... But I doubt that this mathematical permutation test has anything to do with testing what most people's intuition is about. The average spectator, I would think, is saying, man, this guy, Thompson is playing great tonight (in comparison to his usual performance). And, lo and behold, the spectator is right: after scoring a higher percentage of his shots in his opening than usual, he continued scoring well throughout- ending up scoring sixty points in a single game. I would be surprised if the average spectator is saying that because the last three shots were baskets, that the very next one will also be one, but rather, that some nights he scores a smaller percentage, other nights a higher percent. And this seems to be one of those nights that he is playing better.

With a bit more data you can make this more accurate by adding an additional variable because timing is important. In fact, that's one of the main theories about a "hot hand". If someone makes 3 in a row, then you don't give them the ball for 5+ mins or it's half time, maybe they get "cold". Maybe a future study will find out a "hot hand" does exist, but it's limited within one or two minutes in between attempts while the muscle memory is still strong. The study might be something like, if a player makes 3x in a row, is there a statistical significant variance on the 4th shot based on the closeness in time to the third shot? Just a thought.

"Splash brothers" on numberphile! Never thought I'd hear it. Fabulous start to my day.

Some others have touched on this, but the issue I find with these studies is that they don’t define a hot hand in the way I think most basketball fans would. In these studies, a hot streak ends when a player misses a shot, whereas in reality there are plenty of situations in which a player can miss a shot but continue a hot hand after that. For example, a player might be on a streak, miss a shot, get a rebound, and score on that same possession. In this experiment, that player’s hot hand would’ve been deemed over, but most people watching would actually think that the player’s hand is ever hotter at that point! In this way, I think the term “streak” is misused by sports fans, when the term would more appropriately be “hot trend.”

With the given string of 0s and 1s, it is very easy to see that the algorithm (or whatever you wanna call it) would give us a bad rating. The string with the best rating is one starting with 0s and ending with 1s, without anything mixed in, while the worst string is the one where 1s and 0s are mixed together as evenly as possible. The given string already looks pretty even, only at the end it gets a bit strange.

Fantastic analysis. Prof knows her stuff :D

I am sure that in snooker, it makes an enormous difference whether a player has been on the table a long time. It is striking how badly a normally good player tends to play if they have not played a shot for a few minutes

Like a famous German football player said, football is like chess, without rolling dice. I think the numbers are enough evidence to show the effect is minimal, but there are two psychological effects not canceling out: to fear the hot hand opponent on one side and getting confidence in repeated success on the other side. Also, everybody knows feelings of "being in a flow" in any type of work. So there is some intuition about this telling us this is not necessarily wrong. Overall how good a player or team plays is not just random anyway. Therefore I like this kind of precision about how to tackle the statistics to confirm or disprove within an objective context and questioning the objectivity. I like the outlook of how that could have another turn when taking into account the timing.

But this example with Klay's streak doesn't take into account that the defence probably closed him down more, thus making the later shots on average more difficult.

"They're dealing with small, finite samples - and Small stuff does not behave like the big stuff" The mathematical equivalent of the divide between relativity/quantum mechanics?

Only in science and math do you find people with such dedication to truth that they first go through this amount of effort to understand something, only to realize they did it wrong, only to spend additional countless hours to confirm that, despite their original errors, the results of their first efforts were, indeed, correct after all.

wait. but that's not how you accurately interpret data. shouldn't the last 2 be counted twice, since they had 2 times more data than the others. and that would make it exactly 50%

Klay Thompson scored 37 points including 8/8 from 3, which was the most points and 3's ever scored in a single quarter in nba history. He also didn't even play the 4th quarter of that game. For reference his average is only just over 20 a game. When we say a basketball player has a hot hand it is also in reference to a short period of time within that game. It does not last a whole game. What you should be doing is looking at his most average game, take the string from that and compare that to his string from this game. I can guarantee you'll find the sets to be significantly different. On a side note seeing a numberphile video talking about the NBA literally made my day

This is a really super simplistic view on such a complex problem. I understand the need to make assumptions, but to somehow determine 'hot handedness' from a score of 1's and 0's from professional mathematicians only strengthens my view that what is a statistical problem is best solved by statisticians who take into account more variables and their co-dependencies. Mathematicians only seems to be interested in a basic prima faccia view on things...

I love listening to smart scientists setting up their design. Like, as a future nuclear engineer hopeful, experiment design seems like 90% of the challenge.

Doesn't the fact teams are specifically playing to and strategizing for this "hot hand" effect change the results? If a player keeps getting passes, or maybe getting situations set up for them, wouldn't that increase their chances of scoring? Would be interesting to see a comparison between players who were believed to have the "hot hand" versus players who were not and were just shooting without any of that setup.

Surely you could use their fg% and see whether they shot above or below after a streak of shots and misses

With the permutation test, it doesn't take into account that the amount of successful shots could've been different. Think about it, if you were truly to have a hot hand, you would get more total successful shots. Although this does depend on the 'hot hand' being defined not in terms of just the previous X shots being successes, but rather, the idea that successes have a positive feedback loop in terms of overall performance. For instance, if you miss 10 times in a row, you'll feel really bad and discouraged, which will impact your mental state, and cause you to play worse. Whereas if you're doing really well, you feel like you're on top of the world, and you will perform better than average. After all, this is how the hot hand was stated near the start of the video, that you get a better performance than your lifetime(or seasonal) average when you're doing well. It's an important distinction to make, and I do think the "hot hand" as I just described it is a thing. Now back to the permutations test, this definition certainly doesn't allow for an unbiased permutation test. Because the permutation tests are based off a set amount of successes out of a total. So, in a situation where a player got the theoretical hot hand, they would perhaps make 30 out of 40 shots, whereas if they hadn't gone into the hot hand, they'd only make 20 out of 40. No permutation in the test would match the latter case, and it would simply conclude that "he didn't score most of them in a row, therefore he didn't have the hot hand". While my last paragraph was entire a repeat of an earlier sentence, I just hope I make it clear; it very much depends on what you call the hot hand.

The biggest variable these studies seem to ignore is the average defensive ability of the opposing team. So like a hot hand for shooting, a team could also have a streak of blocking shots, stealing balls, etc. Also, what type of shot is not considered, nor the difficulty of a given shot in the pros; for exampke, one can score the same number of points with fewer three point shots than two point ones.

The biggest problem I see with comparing coin flips with basketball shots is the amount of control the player has over the outcome. A great player can have a huge influence on whether his/her shot will land or not, while a coin flipper just has pure luck to count on (or something really really high, given the low amount of control one can have over a coin flip). You can't really compare the two. If you see a great player getting a lot of the shots in, you'll count on the possibility that their body and mind have reached a point while that game is going on, where the player seems to have such control over the shot, that it'll be more likely for the shot to go in. It's called building consistency. If you try to deny that such an effect can happen temporarily during a game, the only logic conclusion you can get after that is that training to build consistency doesn't matter. Because if you argue that one can't improve their consistency (even if temporarily) during a game, you're claiming that practice does not improve your consistency at all, because if it did, even if just a small amount, then the effect would happen during live games as well, which contraditcs your original assumption.

Once you have strong shooting fundamentals down making baskets becomes more subliminal or psychological. If you miss your first 5 shots you're going to be in a different state of mind, like thinking too much about the shot. When you make a lot of shots in a row you just shoot without overthinking it, relying muscle memory, which cause you to have better success when shooting.

Please more Lisa Goldberg!

This takes me back to when I read "The Drunkards Walk" by Leonard Mlodinow.

I understand what they are trying to say, but there are so many other factors that aren't accounted for in this experiment when it comes sports that it's impossible to be accurate.

I like the experimental design, but I’d like to see them incorporate the difficulty of the shots into it. If a player makes 3 3-pointers back to back, that would constitute “being hot,” even if they missed a free throw immediately afterward.

Excellent video!

I LOVE YOU on a more serious note, how well does the sequence comply with the idea that the guy just scores X% of the time?

This may be me just falling into the same fallacy, but I still don't feel like the shots would be independent. "In the zone" to me means two things: some kind of muscle memory is reinforced by getting something in, and therefore doing the same thing as last time may make it easier to score. The second thing is the positive reinforcement, that maybe the belief of streaks being real improves the player's mood in a way that might make it easier to score, just like the little train that could. -I am not a basketball player -I am not a statistician -I am probably talking out of my rear end

In the real world situation, there's a placebo effect, as well as a tendency toward confirmation bias (disregarding data that doesn't support)

Choosing to shuffle the order of the results to me is stupidity. Such an operation assumes no causality, and discounts the psychology of making a shot. Time is not randomly sliced up, it proceeds from the past towards the future. When one makes a shot, it builds confidence and therefore influences future successes. So reording event history to do some sort of statistical analysis seems nonsensical.

I would have thought that at highest level the biggest influence on the outcome of a shot would be all of the regular training and practice, followed by all of the lifetime experience and then followed by any off the field or personal events affecting the player's overall mood on the day. Perhaps at an amateur level where there is less training and other experience the confidence and inherent experience of having scored several prior shots would have a big impact but it shouldn't be as significant for the top pros.

I am a bit confused about the "Fraction of H after H" measure. I really wanted to at that point count the number of H's after H's and divide that by the total number of H's that had something after it. And that would be 4/8. Why do we care about the fraction within the 3 length HT strings? I started getting confused, because what we are asking at that point is confusing to think about: "In all possible HT strings of length three, what is the expected number of H's after H's in an arbitrary HT string of length three?" Though I think typing this may have helped me think about what we are doing. We are looking for the average percentage of shot made after shot (H after H) that occur in HT strings of length three that have at least one shot attempted after an shot. Its still kind of weird to think about though, because of the fact that certain strings are discarded, and that each string doesn't have the same amount of pairs that matter. Awesome video by the way. I really love thinking about my statistical thinking biases.

Has anyone ever studied this using shot quality as a factor? A lot of stats nerds in Hockey spend a lot of time talking about high or low danger shots and passing feeds.