Amanda Cabrera – (no title)

Your essay about issues of authenticity and chance in music and other art forms is mostly very good! You raise some interesting issues and introduce one superb example (the Carlos Santana performance from the 2005 Academy Awards show).

I have marked a number of relatively small corrections on the paper itself. Almost all of these are issues of word choice, or of getting a phrase exactly right. Your writing is generally good, but there are enough of these here that I would say this is something you should think about. I find it often helps to try reading something I am writing out loud (perhaps to another person). Awkwardnesses really show up when you do that.

Specific Comments

1) This is a great example and it shows that at least some composers have such a personal attachment to the music they create that they resist having other musicians reinterpret what they have done. As you go on to say, of course, not all composers think this way though.

2) This raises a couple of very interesting points. First, to what extent did the composer actually have a specific instrument in mind in composing the piece? There are examples both ways, of course. A lot of classical music is very specific about what instruments should be used because the specific quality of the sound the instruments make is part of what the composer had in mind. (Think of the difference between a melody played on a flute and the same melody played on a trumpet!) But there are other examples (such as some pieces that J.S.Bach wrote toward the end of his life) where there are no indications at all what instruments should be used. It almost seems as though what he wrote is “pure music” that could be (and has been) performed on almost any combination of instruments that could produce the notes. Second, to what extent was the composer actually satisfied with what might have been available at the time the music was composed? There are some very well-documented cases where a composer was very dissatisfied with the sound quality of the instruments that were available to him. Beethoven, for example, constantly complained about the weaknesses of the pianos that were available in the early 1800’s, and destroyed several pianos he owned out of frustration (even before his deafness!). It seems very likely he would have jumped at the chance to have his music performed on a modern grand piano – it would have been closer to the “ideal instrument” he heard in his mind’s ear.

3) It is also interesting to realize that, at the time much “classical music” was written, it was essentially the popular music of the time. The fact that it has survived so long creates whole new layers of questions about authenticity and interpretation. I doubt that many composers from the second half of the 18th century (other than Mozart and Haydn, perhaps) would have expected that their music would still be performed over 200 years later!

Steph Craig – The Underlying Meanings of Music

Good job on this assignment. Your writing is really strong, and you raise some interesting points. However, I think one of your basic working assumptions is somewhat questionable (even though it is certainly intuitively appealing). See comment 1 below for this.

Specific Comments

1) The sentence I marked here does not seem to follow what you said before. I thought you were going to say “It is possible however that these messages are not the ones that listeners perceive.” And this whole paragraph raises an interesting and important point about music. Namely, I agree that when you are listening to a piece of music, it can certainly seem that some sort of information, or emotional state, or story is being conveyed. But if you try to pin down exactly what that is, it is surprisingly hard. Moreover, it is never clear that different listeners will “get” the same thing from a given piece of music if they are presented the music “cold,” that is, without any accompanying verbal explanations or cultural associations. Many philosophicallyinclined musicians who have tried to think about these questions have come eventually to the point of view that there is no underlying meaning in music at all – just patterns of sound that provoke different responses in different listeners. (You actually come around to saying something like this on page 4.) That is somewhat unsatisfying, but if you want to argue that there is meaning in the music itself, then as I said before, it is hard to specify exactly what it is!

2) By “their attempts are justified,” I think you are trying to say something more like “they can try to control the perception of their music if they want (by specifying how it should be performed).” But as you say next, this kind of attempt is essentially doomed to fail because performers will do their own thing anyway, and individual listeners will always interpret the music they hear on their own terms. I more or less agree with what you are saying here as a practical matter. Your “everything’s OK” position is a demonstration of what I was saying above about the difficulty of identifying exactly what is conveyed by music, though.

3) I think another aspect of the “tortured feeling” you bring up here is that it is often extremely difficult for composers (or writers, or artists, ... ) to realize their artistic vision in their chosen medium. Having a great idea for a piece of music is the first step, but actually getting it down on paper, and then having it performed require struggles that can often “eat at a composer’s soul,” as you say.

Dan Schmader – Artists, Choices, and Chance

This is a good paper on some of the issues about authenticity, originality, etc. in performance of music. You have some interesting and different viewpoints about some things and you have expressed them well for the most part. The major criticism I have about your writing here is that the organization into paragraphs is not always very clear. Paragraphs seem to run on very often even though you are introducing new ideas or changing topics. The conclusion is also a bit weak because the last paragraph repeats many points you made before without developing them.

1) This is certainly true. Being remembered and having one’s music played, even if it is played in different ways than one expected in one’s lifetime, would certainly be preferable to a composer than being forgotten entirely!

2) This is an interesting idea – if I understand you, you are saying that by bringing their own knowledge of the modern instruments into play, Carol and Mark were adding something of themselves (or their own time) to the music and were more original than when they were playing the baroque instruments because of that. I never thought of it this way, but you might have a point, if you are thinking of the personal contribution of the performers when a piece of music is played and heard.

3) I do not know what you were getting at here. What did you mean when you said that Beethoven was gaining authenticity? (Also, you have a very long paragraph here that addresses a number of different, and not too closely related points. It should be divided into several smaller paragraphs, for instance by starting a new paragraph where I marked.)

4) You are right that Mozart contributed a great deal to the evolution of music over his lifetime. In addition to the example you give, it was also true that he was a leader in providing more prominent and more interesting and idiomatic music for woodwind instruments (clarinet, oboe, bassoon, French horn especially, flute to a lesser extent).

5) I know exactly what you mean here about the unexpected or unplanned things that can happen in musical performances. But I don’t think these chance occurrences are necessarily a case of “error.” The might be different from what the composer put down in the score, but they are not strictly wrong! Also, I don’t think your example the way Mark Kroll tried to get the audience to be really quiet or the way the lighting was set up in the room are really good examples of allowing for chance occurences. (It seems to me that, if anything, they are exactly the opposite – attempts to minimize the effects of randomness by getting the audience to be really still and to focus their attention on the stage area!)

6) Your last paragraph is another very long one, and it is mostly a “rehash” of ideas you had brought up earlier. A strong conclusion adds a lot to a paper (and a paper slightly shorter than 5 full pages would be OK provided that the conclusion really brought things to strong close!)

Shelby Proft – Artists, Choices, and Chance

I am not sure I agree with what you are saying about randomness in music, though the difference is one of degree, perhaps. I would say music performance is a mixture of random and deterministic elements instead of being “completely random.” Nevertheless you have used a good example and you have some interesting points.

I think you are also making good progress on the writing front. This paper is much clearer and more understandable than your work early last semester. There are still a few places where some rewording would help. But in general, this is good work!

Specific Comments

1) I don’t quite see how the differences in the sounds of the harpsichord and the modern piano are really a product of randomness. The piano was invented by an 18th century instrument maker who was trying to provide an instrument that could do different things than a harpsichord could. That is, pianos were designed to play loudly and softly depending on how the keys were struck, and also to be capable of sustaining sounds for a longer time than the harpsichord could. This was all very intentional, not random.

2) The human voice is the original musical instrument (for humans anyway!) and the ideal toward which almost all instruments strive. (For example, we say that other instruments can “sing” if they are played skillfully enough.) Also, you might be interested to know that I have read that the use of auto-tuning actually started out as a way to correct spontaneous mistakes in pitch made by singers and make their live performances sound more like their recordings. (To me this seems to be hardly different from lip-synching.) On the other hand, there are apparently some singers who make it a point not to use the auto-tuning machines for that purpose, because they think their authentic voices, with their small imperfections, are more expressive. So this kind of technology can cut both ways.

3) This is possible, but there are also some pieces that J.S.Bach wrote toward the end of his life where no indications at all are given what instruments should be used. It seems as though what he wrote was “pure music” that could be (and has been) performed on almost any combination of instruments that could produce the notes.

4) Again, I’m not sure if I agree with what you are saying here about the “completely random” nature of music – this seems too strong and too sweeping a statement. As you say, the performance of music certainly contains random elements. But there are nonrandom aspects too if the performers are trying to reproduce the music described by the instructions in a musical score.

5) It took me a while to figure out what you meant to say here, but I think this is an interesting point. If I understand what you meant by “continual,” it is that there would only be one completely authentic performance that would be repeated over and over with no changes. That would be boring!

Jonathan Ye – Does Randomness Exist?

This is a very well-written and entertaining essay. You have made (quite slyly and indirectly, but very effectively) an interesting point about the connection between apparent randomness and the information we have available. Things often look random because we don’t have complete information about them, and on the other hand things can be said to be random if they have a high degree of complexity or information content (see comment 1 below).

The only real criticism I have is that you might have included a somewhat fuller and more mathematical explanation of chaos in dynamical systems – see comment 2 below. Also, you are right that radioactive decay is an example of a physical process that is considered to be truly random (according to a standard interpretation of quantum mechanics). Of course, it is also possible to ask whether it even makes sense to consider an algorithm that would predict the decay of an atom of a radioactive isotope of some element like U^{235}! Is there anything that says such an algorithm must exist? In other words, how good is the proposed idea of randomness that you describe?

Specific Comments

1) I assume this is the MIT student Qiaochu Yuan whose blog I found via Google(!) You might be interested to know that this proposed definition of randomness is a version of an idea called the Kolmogorov complexity. In theoretical computer science, the Kolmogorov complexity of a string is the length of the shortest program that generates that string (where the program is in some universal language powerful enough to specify any algorithm). High complexity means that the shortest program that generates the string is comparable in length to the string itself. For instance the string “ababab ⋯ ab” with two million alternating copies of the characters “a”, “b” has very low complexity since a program to generate it could say, in effect, “repeat “ab” one million times,” and this is far shorter than 2 million characters long(!) This is one way to understand the fact that the string is not random. On the other hand, there might not be any way to generate a “really random” string shorter than just listing all the characters that make up the full string. This seems like a reasonable intuitive operational definition of randomness to me, at least in certain domains. For instance, it might be more applicable to abstract mathematical objects than to physical processes like radioactive decay(!)

2) Not exactly. Chaos theory is really the mathematical discipline that studies these phenomena. And a situation as simple as a dart in free fall is not one where “chaotic behavior” would be expected. In a chaotic dynamical system, such as the Lorentz system of differential equations, in principle, a trajectory is uniquely determined given the initial conditions. However, because of the sensitive dependence on the initial conditions, it is necessary to know the initial conditions exactly in order to make predictions about the long-term behavior of the trajectory. The initial values we need to know are represented mathematically as real numbers with infinite and nonrepeating decimal expansions. We can only make measurements to some finite precision (some finite number of decimal places). So even though the system is deterministic in principle, prediction is not possible because of the fact that we will never know the initial conditions exactly.

3) Pedantic comment – togas were worn in ancient Rome, not Greece. Although some ancient Greeks wore something similar, they were called by a different name(!)

Kyle Walraven – On Chaos and Randomness

This is an excellent and very well-written essay on a very subtle topic. I think you have the makings of a philosopher(!)

The only major critical comment I have is that your proposed definition of randomness has some deficiencies. (But don’t feel bad because this is an area where philosophers, mathematicians, and scientists have yet to come to complete agreement!) See comment 1 below for a first objection. And see comment 2 for an example that shows you need to be careful about the idea of “unpredictability” too. Does this mean “unpredictability in practice” or “unpredictability in principle?” There’s a difference, and there are deterministic (i.e. non-random) systems that are also unpredictable in practice.

You are right to point out that things often look random because we don’t have complete information about them. On the other hand it is interesting to note that one provisional definition of randomness (different from the idea you were trying to get across) is that randomness is essentially the same thing as a high degree of complexity or “information content.” There is another proposed definition of randomness coming from what is called the Kolmogorov complexity in theoretical computer science. The Kolmogorov complexity of a string is the length of the shortest program that generates that string (where the program is in some universal language powerful enough to specify any algorithm). High complexity means that the shortest program that generates the string is comparable in length to the string itself. For instance the string “ababab ⋯ ab” with two million alternating copies of the characters “a”, “b” has very low complexity since a program to generate it could say, in effect, “repeat “ab” one million times,” and this is far shorter than 2 million characters long(!) This is one way to understand the fact that this string is clearly not random. On the other hand, there might not be any way to generate a “really random” string shorter than just listing all the characters that make up the full string. This seems like a reasonable intuitive operational definition of randomness to me, at least in certain domains. But for instance, it might be more applicable to abstract mathematical objects than to physical processes like radioactive decay(!)

Specific Comments

1) This seems to be much too strong a requirement to me. A strict scientific materialist (one who believes that there is nothing else besides matter and equivalent energy in the universe – no “spirit” or consciousness independent of physical processes in matter) would argue that everything has to be governed by natural laws. It seems to me that your other ideas should be consistent with that position, so you don’t want to rule out that possibility with how you define randomness. Moreover, there are physical theories (such as a standard interpretation of the theory of quantum mechanics that describes matter on the atomic and subatomic level) where randomness is “built-in” in a certain sense. In fact, certain physical processes described by quantum mechanics, such as radioactive decay are usually considered to be purely random. Of course, like all scientific theories, quantum mechanics is just the best description we know at present of those physical phenomena, and it could turn out that in the future a different understanding, eliminating the randomness, will be achieved(!)

2) There is a fairly subtle point that could be better explained here. In a chaotic dynamical system, such as the Lorentz system of differential equations in meteorology (where the “butterfly effect” was discovered), in principle, a trajectory is uniquely determined given the initial conditions. So in mathematical terms, the system is strictly deterministic(!) However, because of the sensitive dependence of the trajectory on the initial conditions (and the strong mixing properties of the Lorentz system), it is necessary to know the initial conditions exactly in order to make predictions about the long-term behavior of the trajectory. But the initial values we need to know are represented mathematically as real numbers with potentially infinite nonrepeating decimal expansions, and we can only make measurements to some finite precision (some finite number of decimal places). So even though the system is deterministic in principle, prediction is not possible because of the fact that we will never know the initial conditions exactly. In other words, it is possible for a system to be predictable in principle, but unpredictable in practice.

John Vatalaro – Streaks and Statistics

John, your paper on streaks and statistics is well-written and you make your points strongly. However, I think that some of your reactions to Mlodinow’s contentions were maybe colored a bit too much by emotion (see comment 2). I don’t see that Mlodinow was trying to say that DiMaggio’s streak is any less admirable just because similar or longer streaks are actually somewhat likely on the basis of probabilities. I think he is just trying to explain that it may not be as exceptional as we would like to think it is. It is also true that there are certainly questions one could raise about the specific ways that the 10,000 simulations of baseball were carried out.

Specific Comments

1) I don’t quite agree that this is what Mlodinow was saying. I think it would be fairer to say that his point was that in any sport (and even more generally in many areas of life), streaks of positive or negative outcomes can happen, due to chance, more often than our human perceptions are geared to understand. He would be the first one to acknowledge that long streaks of positive outcomes will happen more often for players who have a high level of skill or dedication than for others – their skill and dedication makes it more likely that a good outcome happens in every at-bat, so it is also more likely that they have longer hot streaks. Note that the simulated players who beat DiMaggio’s record the most often were not “slouches;” they were other acknowledged major stars like Ty Cobb(!) Whatever you think of Cobb as a person (he was apparently a really unpleasant character in several ways), he was certainly one of the all-time greats in the game.

2) I guess it is natural to react this way, especially if you see the conclusion of this article as a threat to the memory or reputation of a great player like DiMaggio. However, again I would say that this is not Mlodinow’s real point. I think the basic thing he is trying to get across is that streaks are more common and longer than we tend to think based on our limited understanding of probabilities. I don’t think that takes anything away from DiMaggio, but it does raise the question why his record has stood for so long(!) And if you want to really argue the position you are taking here, I think it would be good to try to come to a more explicit explanation of exactly what it is about the full human experience of playing a sport that makes a box model an insufficient representation of the “real thing.” Here is one example of a real issue I see – it may very well be the case that using a single life-time batting average in a simulation like this “smooths things out too much” and ignores the up and down variation in a player’s performance over a single season. Running into a local cold stretch over a couple of games could easily “kill” a hitting streak. I have not looked into the details of how Arbesman and Strogatz carried out their simulation enough to be able to say whether they anticipated this criticism and designed their computer program accordingly. I think it would be very interesting to learn more(!)

Brandon Nunn – How likely are “streaks” in sports?

This is a generally well-written essay on the points Mlodinow raised in his article about streaks in general and DiMaggio’s streak in particular. I sense that you may be “of two minds” about this, though, because the first three pages of what you wrote seemed to be heading toward saying that the simulation of 10,000 alternate baseball histories by Arbesman and Strogatz might not have been realistic enough to justify the conclusion that streaks like DiMaggio’s are not that uncommon. But then you spend the rest of the paper more or less summarizing what Mlodinow said in The Drunkard’s Walk about some reasons why human perceptions are not well developed for the estimation of probabilities. The ultimate conclusion is good, but the level of mis-estimation of probability involved in Arbesman and Strogatz’s model strikes me as much less than the crude estimates of probabilities that humans make naively. I guess you are trying to say that even people using the mathematics in a correct way can come to incorrect conclusions if they don’t capture all the relevant aspects of the real world in their mathematical models. It might have been better to make this point even more explicitly.

Specific Comments

1) I don’t quite agree that this is what Mlodinow was saying. I think it would be fairer to say that his point was something you come around to saying later in the paper: In any sport (and even more generally in many areas of life), streaks of positive or negative outcomes can happen, due to chance, more often than our human perceptions are geared to understand. He would be the first one to acknowledge that streaks of positive outcomes will happen more often for players who have a high level of skill or dedication than for others – their skill and dedication makes it more likely that a good outcome happens in every at-bat, so it is also more likely that they have longer hot streaks. Note that the simulated players who beat DiMaggio’s record the most often were not “slouches;” they were other acknowledged major stars like Ty Cobb(!) Whatever you think of Cobb as a person (he was apparently a really unpleasant character in multiple ways), he was certainly one of the all-time greats in the game.

2) You are raising some very good points here – it may very well be the case that using a single life-time batting average in a simulation like this “smooths things out too much” and ignores the up and down variation in a player’s performance over a single season. Running into a local cold stretch over a couple of games could easily “kill” a hitting streak. I have not looked into the details of how Arbesman and Strogatz carried out their simulation enough to be able to say whether they anticipated this criticism and designed their computer program accordingly. I think it would be very interesting to learn more(!)

3) I think you could say the opposite, though. Aren’t those human qualities exactly what would lead a player to accumulate a .325 lifetime average in the first place? No one who wasn’t very talented, persistent, and extremely dedicated could do this over a 15-year career.

4) You are starting to overuse the superlatives like “incredible” at this point! Ripken’s consecutive game streak strikes me as a very interesting combination of a very tough mental attitude that allowed him to play through the minor aches and pains all baseball players have, plus a lot of luck in that he never had a major (say season-ending) injury through those years. The times I saw him play, I certainly never got the impression he was “playing it safe” or holding anything back. Maybe his body was more durable than average, but a lot of this record has to be attributed to luck, I think.

Nick Foster – Joe DiMaggio’s Streak

You argue your position (that probabilities are not the last word in human activities like sports and that they should not be applied to questions such as whether a streak like DiMaggio’s is likely) quite strongly and eloquently. Your writing is also generally very good. I’m not sure I would go as far, though. I would not totally rule out the possibility of gaining insight about the outcome of sporting events by using probability-based simulations like the one by Arbesman and Strogatz that Mlodinow describes, even though, as you point out, it is not entirely clear to what extent the results of that simulation should be believed.

Specific Comments

1) I don’t quite agree that this is what Mlodinow was saying. I think it would be fairer to say that his point was that in any sport (and even more generally in many areas of life), streaks of positive or negative outcomes can happen, due to chance, more often than our human perceptions are geared to understand. He would be the first one to acknowledge that long streaks of positive outcomes will happen more often for players who have a high level of skill or dedication than for others – their skill and dedication makes it more likely that a good outcome happens in every at-bat, so it is also more likely that they have longer hot streaks. Note that the simulated players who beat DiMaggio’s record the most often were not “slouches;” they were other acknowledged major stars like Ty Cobb(!) Whatever you think of Cobb as a person (he was apparently a really unpleasant character in several ways), he was certainly one of the all-time greats in the game.

2) I agree that there are some questions one should ask before accepting the results of this simulation of 10,000 histories of baseball. For instance, did they really use a single lifetime batting average for each player (effectively treating each player like a robot)? I think it is clear that doing the simulation that way would “smooth things out too much” and ignore the up and down variation in a player’s performance over a single season. Running into a local cold stretch over a couple of games could easily “kill” a hitting streak. I have not looked into the details of how Arbesman and Strogatz carried out their simulation enough to be able to say whether they anticipated this criticism and designed their computer program accordingly. I think it would be very interesting to learn more(!)

3) Likewise, I don’t know the extent to which Arbesman and Strogatz tried to take into account the head-to-head matchups of batter versus pitcher in their simulated histories. I think that doing this in a realistic way might address the objection your are raising here. Programming a simulation with that level of detail would be difficult, but I don’t see that it would be impossible(!)

4) It is certainly true that the unlikely “upset” outcomes are the ones we remember the most and they are a big part of why people follow sports. But is it really wrong to ask how unlikely a given feat really was? It seems to me that an informed understanding of exactly how likely or unlikely some favorite sports record was is something a true fan should want to have!

5) Picky point – these are not complete sentences. You could say something like: “Consider the USA-Soviet hockey game from the 1980 Olympics, ... ”

Carl Losito – How Likely are “Streaks” in Sports?

I agree that your position (that skill and dedication have to count for something in sports) is intuitively appealing. But also see comment 7 below for one possible response to this. Of course, there are also questions about the way the simulations of 10,000 alternate histories of baseball were carried out that would have to be addressed in order for those conclusions to carry the amount of weight that Mlodinow attributes to them (see comment 5). But I think you probably would agree that this certainly thought-provoking!

I think your writing has improved over the year, but there are still a number of places where your word choices seem obscure to me. The specific terms you use are hard to understand because it is not obvious that they fit with the rest of what you are saying. I marked a number of these in the paper, and see some of the comments below as well. Another technical point is that in an extended discussion of an article like the one you are doing here, it is certainly OK (and it sounds somewhat more natural to me) to use the present tense consistently – say Mlodinow introduces, concludes, describes, etc. rather than introduced, concluded, described, etc.

Specific Comments

1) I don’t quite understand what you are getting at here. To “stifle” someone usually means to keep him or her from speaking or acting by exerting some type of influence or force. I can’t see how that applies here.

2) I think you mean something like “can come about through chance.”

3) This is not quite correct. Mlodinow reports the results of the simulated 10,000 alternate histories of baseball, but he did not carry out the simulations himself. They were done by two mathematicians at Cornell, Arbesman and Strogatz. Be careful about attributions of credit like this.

4) Again, I don’t quite understand what you are getting at here. How does denying the influence of randomness “maintain stability?”

5) I agree that there are some questions one should ask before accepting the results of this simulation of 10,000 histories of baseball completely. For instance, did Arbesman and Strogatz really use a single lifetime batting average for each player (effectively treating each player like a perfectly consistent robot)? I think it is clear that doing the simulation that way would “smooth things out too much” and ignore the up and down variation in a player’s performance from day to day over a single season. Running into a local cold stretch over a couple of games could easily “kill” a hitting streak. I have not looked into the details of how Arbesman and Strogatz carried out their simulation enough to be able to say whether they anticipated this criticism and designed their computer program accordingly. I think it would be very interesting to learn more(!)

6) I don’t quite agree that this is what Mlodinow was saying. I think it would be fairer to say that his point was that in any sport (and even more generally in many areas of life), streaks of positive or negative outcomes can happen, due to chance, more often than our human perceptions are geared to understand. So the likelihood that someone would have had a similar streak is high. It is not possible to say with certainty that someone else would have done it.

7) I think it is important to understand that Mlodinow would definitely be the first one to acknowledge that long streaks of positive outcomes will happen more often for players who have a high level of skill or dedication than for others – their skill and dedication makes it more likely that a good outcome happens in every at-bat, so it is also more likely that they have longer hot streaks. Note that the simulated players who beat DiMaggio’s record the most often were not “slouches;” they were other acknowledged major stars like Ty Cobb(!) Whatever you think of Cobb as a person (he was apparently a really unpleasant character in several ways), he was certainly one of the all-time greats in the game.

Josh Hauser – Streaks

You raise some good points in your essay about Mlodinow’s article on DiMaggio’s 56game consecutive hitting streak. But I don’t think your ideas carry the weight they could because of the way they are presented. See the comments below and some corrections marked on the paper for some suggestions on issues related to the writing. In general, I would say that you are using a somewhat too informal style here. Phrases like “ridiculous season” are fine in everyday speech, but you should try to find less colloquial equivalents for formal writing. Also, in referring to the author of a book or article you are discussing, you should always use the person’s last name (i.e. “Mlodinow”) not the person’s first name (i.e. “Leonard”). You are not on a first-name basis with him here, so to speak(!)

Specific Comments

1) You should say here that the simulation you are referring to was conducted by two mathematicians at Cornell (Arbesman and Strogatz).

2) I think everyone would agree that there was some luck or randomness involved in DiMaggio’s streak. To say otherwise is to say, in effect, that he could control every aspect of the games he played in. If that was true, why did the streak end? The real issue is how likely it was for anyone to have a consecutive game hit streak that long or longer.

3) It would be a nice exercise to try to see where this 75% chance figure came from. Think about the binomial probabilities for getting at least one hit in 4 at bats, if you use DiMaggio’s lifetime average of .325 as the probability of getting a hit in any one at-bat.

4) I don’t think that is quite what the article says. The point is that people tend to think of “hot” or “cold” streaks as coming from some change in the athlete. Saying you are “in the zone,” for instance seems to say that you have found some particularly productive mental state to be in, and that is what is causing the better than average outcome. But what the article is arguing is that the same kind of hot or cold streaks would be observed if there were no changes in the athlete, just normal random variation in the outcomes.

5) You are raising a very good point here – it may very well be the case that using a single life-time batting average in a simulation like this “smooths things out too much” and ignores the up and down variation in a player’s performance over a game or a single season. Running into a local cold stretch over a couple of games could easily “kill” a hitting streak. I have not looked into the details of how Arbesman and Strogatz carried out their simulation enough to be able to say whether they anticipated this criticism and designed their computer program accordingly. I think it would be very interesting to learn more(!)

6) This is another very good point. I think this says that in effect the Yankees’ opponents did not make any special attempt to shorten the streak – they may have thought that, out of a spirit of good sportsmanship or whatever, they should give DiMaggio a chance to keep the streak going if he could.

7) Careful – these factors are not “confounders” in the precise sense we used that term in the first semester. They are not factors associated with the treatment and the outcome of a study.

8) You seem to be arguing against your own point here. With 30 teams, there are many more players active in the major leagues than there were when DiMaggio was playing (and there were only 16 teams). This should make it more likely that a long hitting streak would happen now.

9) I don’t know about this either. You could certainly argue, though, that with the increased specialization of starters with strict pitch count limits, set-up men, eighthinning pitchers, and closers, batters are facing consistently fresher pitching.

Patrick Jarvis – How likely are streaks?

This is a good essay about Mlodinow’s discussion of streaks in baseball, although there are some places where I think you are misrepresenting what Mlodinow was saying–it is a bit more “nuanced” than you are acknowledging. It was good that you tried to explain some of the probability calculations. But there was one small thing that you overlooked. See comment 5 below for the correct way to estimate the probability that a weaker team will win a 7-game World Series in the particular case you were looking at.

See the paper for some (relatively minor) suggestions for rewording or rephrasing some things.

Specific Comments

1) I don’t quite agree that this is what Mlodinow was saying. I think it would be fairer to say that his point was that in any sport (and even more generally in many areas of life), streaks of positive or negative outcomes can happen, due to chance, more often than our human perceptions are geared to understand.

2) Mlodinow would be the first one to acknowledge that long streaks of positive outcomes will happen more often for players who have a high level of skill or dedication than for others – their skill and dedication makes it more likely that a good outcome happens in every at-bat, so it is also more likely that they have longer hot streaks. Note that the simulated players who beat DiMaggio’s record the most often were not “slouches;” they were other acknowledged major stars like Ty Cobb(!) Whatever you think of Cobb as a person (he was apparently a really unpleasant character in several ways), he was certainly one of the all-time greats in the game.

3) I don’t understand what you are trying to get at here. The issue is not really “accuracy of results.” It is how likely the “outliers” are.

4) This figure depends on the probabilities that each team wins in a single game – the 40% is based on the 55% − 45% probabilities.

5) This is a good try, but you are not thinking about how to compute the probabilities in exactly the right way. The chance of an event cannot come out to be greater than 100%. The reason one of yours did was that you were counting the same outcome multiple times. Here is the way you need to do it: If the weaker term wins in four games, then the binomial probability formula gives (.45)^{4} (about 4%) as you say. If they win in 5 games, then the thing you were not seeing is that they have to win the fifth game, so the binomial probabilities only apply to the first four games: \binom{4}{3}(.45)^{3}(.55)(.45) (where the last .45 represents the fifth game). There is about a 9% chance of this. Similarly, the probability that they win in six games is \binom{5}{3}(.45)^{3}(.55)^{2}(.45) (about a 12.4% chance), and the probability that they win in seven games is \binom{6}{3}(.45)^{3}(.55)^{3}(.45) (about a 13.6% chance). The sum of these is 4 + 9 + 12.6 + 13.4 ≐ 39% (a bit more, since I rounded down). This is where the “about 40%” figure comes from.

6) “Recreated 10,000” times is not the best way to describe what happens in simulations like this. What the program did was to run through a simulation of the history of baseball 10,000 times, but with different outcomes based on random factors determining the outcome of at-bats, games, etc.

7) Careful – these factors are not “confounders” in the precise sense we used that term in the first semester. They are not factors associated with the treatment and the outcome of a study.

8) I do not see that this is relevant. The issue is how often we should expect a professional baseball player to have a streak like DiMaggio’s. What average people could do is not the point.

9) Again, I don’t think this is what the program that Arbesman and Strogatz wrote was trying to do. They were’nt trying to predict outcomes, just explore the range of different outcomes that were probable enough to show up in the 10,000 simulated histories of the game.

10) You are raising a very good point here – it may very well be the case that using a single life-time batting average in a simulation like this “smooths things out too much” and ignores the up and down variation in a player’s performance over a single season. Running into a local cold stretch over a couple of games could easily “kill” a hitting streak. I have not looked into the details of how Arbesman and Strogatz carried out their simulation enough to be able to say whether they anticipated this criticism and designed their computer program accordingly. I think it would be very interesting to learn more about exactly what they did(!) In other words, while their results are suggestive, I would not want to accept their claims uncritically without more precise information about exactly what they did!

11) It is interesting that you are trying to argue this way, because it seems to me that you could make exactly the same criticism of someone who says that DiMaggio’s streak could only have come about because of the exceptional talent and dedication of a great player. This is the main point of Mlodinow’s article and the Arbesman-Strogatz simulation study too. We underestimate the potential for long streaks just based on chance because we want to believe that humans have ultimate control over the events in their lives.

Greg Monfette – Streaks in Sports

Good essay on the Mlodinow article about the role of streaks in sports. You have brought in some good additional examples and your writing is very strong here. I have marked some minor suggestions on the paper itself, but overall I think this is probably your best writing so for our class(!)

Specific Comments

1) I don’t quite agree that this is what Mlodinow was saying. I think it would be fairer to say that his point was that in any sport (and even more generally in many areas of life), streaks of positive or negative outcomes can happen, due to chance, more often than our human perceptions are geared to understand.

2) As you indicate, Mlodinow would be the first one to acknowledge that long streaks of positive outcomes will happen more often for players who have a high level of skill or dedication than for others – their skill and dedication makes it more likely that a good outcome happens in every at-bat, so it is also more likely that they have longer hot streaks. Note that the simulated players who beat DiMaggio’s record the most often were not “slouches;” they were other acknowledged major stars like Ty Cobb(!) Whatever you think of Cobb as a person (he was apparently a really unpleasant character in several ways), he was certainly one of the all-time greats in the game.

3) Instead of saying “world/dimension” here, it might be better to say “simulated history.”

4) You are raising a very good point here – it may very well be the case that using a single life-time batting average in a simulation like this “smooths things out too much” and ignores the up and down variation in a player’s performance over a single season. Running into a local cold stretch over a couple of games could easily “kill” a hitting streak. I have not looked into the details of how Arbesman and Strogatz carried out their simulation enough to be able to say whether they anticipated this criticism and designed their computer program accordingly. I think it would be very interesting to learn more about exactly what they did(!) In other words, while their results are suggestive, I would not want to accept their claims uncritically without more precise information about exactly what they did!

5) “A better life” sounds somewhat “wishy-washy.” You are saying something much more specific, namely that an understanding of the role of chance would give people a more realistic view of the things that happen in their lives.

Jack Karlovec – Streaks in Sports

You take a pretty strong position here that probabilities basically don’t have much to say about outcomes of sporting events and the likelihood of streaks like DiMaggio’s. I agree that this position is intuitively appealing. Your writing is also very good for the most part – certainly the best writing I have seen from you all year.

However, there is a major aspect of the Mlodinow article that you have not mentioned at all – the results of the simulation of 10,000 alternate histories of baseball by the Cornell mathematicians Arbesman and Strogatz. It seems to me that that deserves some mention, because it would be possible to argue (as Mlodinow does) that those results show the exact opposite of what you are claiming.

Specific Comments

1) I think you are misrepresenting Mlodinow’s point here by saying the alternative is “streaks are simply flukes.” I think it would be fairer to say that his point was that in any sport (and even more generally in many areas of life), streaks of positive or negative outcomes can happen, due to chance, more often than our human perceptions are geared to understand. Mlodinow would be the first one to acknowledge that long streaks of positive outcomes will happen more often for players who have a high level of skill or dedication than for others – their skill and dedication makes it more likely that a good outcome happens in every at-bat, so it is also more likely that they have longer hot streaks. I think it is very telling that the players (in the computer simulations you did not mention) who beat DiMaggio’s record the most often were not “slouches;” they were other acknowledged major stars like Ty Cobb(!) Whatever you think of Cobb as a person (he was apparently a really unpleasant character in several ways), he was certainly one of the all-time greats in the game, with a level of skill and dedication similar to DiMaggio’s.

2) I don’t think that the “hot hand” in basketball is something that everyone would agree about. Mlodinow, in fact, would say there is no such thing. His point is that people tend to think of “hot” or “cold” streaks as coming from some change in the athlete. Saying you are “in the zone,” for instance, seems to say that you have found some particularly productive mental state to be in, and that is what is causing the better than average outcome. But what the article is arguing is that the same kind of hot or cold streaks would be observed if there were no changes in the athlete, just normal random variation in the outcomes due to chance and probability.

3) Bringing up all the circumstances and events that can influence the outcome of baseball games is not a very strong argument against Mlodinow’s position, in my opinion. The reason is that someone who wanted to take the opposite position could just say that all those circumstances are exactly what goes into determining the statistics of players. And it is those career statistics that were used by Arbesman and Strogatz to predict that long hitting streaks should happen relatively more frequently than we want to believe. A much stronger question to raise would concern the way those simulations were done. For instance, it is not very clear from what Mlodinow wrote whether a batter’s lifetime average was used to estimate the probability of outcomes in every simulated at-bat. As you can probably tell, it may very well be the case that using a single life-time batting average in a simulation like this “smooths things out too much” and ignores the up and down variation in a player’s performance from day to day over a single season. Running into a local cold stretch over a couple of games could easily “kill” a hitting streak. Similarly, it is not clear whether the simulation took specific pitcher-batter matchups into account. I have not looked into the details of how Arbesman and Strogatz carried out their simulation enough to be able to say whether they anticipated these criticisms and designed their computer program accordingly. I think it would be very interesting to learn more about exactly what they did(!) In other words, while their results are suggestive, I would not want to accept their claims uncritically without more precise information about exactly what they did!

Mary Arnold – Streaks in Sports

This is an excellent summary of the Mlodinow’s article on the DiMaggio streak and related topics from The Drunkard’s Walk. You were actually the only one who wrote on this topic to more or less acknowledge the possibility that chance could have had a big role in a streak like DiMaggio’s famous example. Everyone else really wanted to hold to the idea that only the human factors you mention should be taken into account! Everyone else seemed to think that saying chance played a role was somehow diminishing DiMaggio’s accomplishment. But of course, Mlodinow would be the first one to acknowledge that long streaks of positive outcomes will happen more often for players who have a high level of skill or dedication than for others – their skill and dedication makes it more likely that a good outcome happens in every at-bat, so it is also more likely that they have longer hot streaks. I think it is very telling that the players in the computer simulations who beat DiMaggio’s record the most often were not “slouches;” they were other acknowledged major stars like Ty Cobb(!) Whatever you think of Cobb as a person (he was apparently a really unpleasant character in several ways), he was certainly one of the all-time greats in the game, with a level of skill and dedication similar to DiMaggio’s.

Your writing is very good – I had only a few suggestions for changes.

Specific Comments

1) When you say it this way, the probabilities of streaks do sound more likely. There is a point here, though, that should not be passed over too lightly. It may very well be the case that using a single life-time batting average in a simulation like this “tends to smooth things out too much” and ignores the inevitable up and down variation in a player’s performance over a game or a single season. This is especially true for the specific situation of the consecutive game hitting streak because one “off day” could easily “kill” a hitting streak. I have not looked into the details of how Arbesman and Strogatz carried out their simulation enough to be able to say whether they anticipated this criticism and designed their computer program accordingly (perhaps by allowing some random up and down variation in the simulated players’ averages over a simulated season). I think it would be very interesting to learn more(!)

2) I think there is also an element of psychology here. To a player in the middle of a hot streak, there can be a perception of being invincible, or “in a zone,” or having a “hot hand.” To them, it feels as though there is something different about them at that time, or that they are doing something different to make the streak happen. I think this is another aspect of our human tendency to look for reasons and explanations of things in our experience.

3) See the general comments above. If you admit that streaks are more likely than we sometimes think because of chance, then it is perhaps not surprising that DiMaggio was the player to have the longest streak in the actual history of the game, because of his high level of talent and consistency.

Paige Carr – Human Misperception of Randomness

This is a good summary of the discussion of why humans tend to misperceive the effects of randomness, along the lines of our class discussions and The Drunkard’s Walk. But this paper was supposed to be somewhat more specific about the probability of long consecutive gave hitting streaks in baseball along the lines of DiMaggio’s famous record streak.

Unfortunately, you have not discussed one of the most interesting aspects of the Mlodinow article from the Wall Street Journal about DiMaggio’s streak at all. I think this means that you have not really addressed some of the interesting features of this particular baseball record, and a key piece of evidence of how likely long hitting streaks might be. I am referring to the computer simulation of 10,000 alternate histories of baseball by the two mathematicians Arbesman and Strogatz at Cornell using the actual players’ statistics for each year to estimate probabilities for different occurrences. Although I think there are questions one could ask about the methods they used, I think their results seem to be quite convincing about the likelihood of long hitting streaks – in 42% of the “parallel histories,” in fact, there were streaks as long or longer than DiMaggio’s actual record streak.

Of course as you mention at one point, the skill and dedication of players matter too for this, and in fact it was usually some of the best players in history (for example, Ty Cobb) who were the ones who had the long simulated hitting streaks.

Your writing is very good here – there were hardly any changes I found to suggest.

Specific Comments

1) In this case, it would still be the case that the expected number of heads or tails would be half the total number of tosses. The complicated psychological thing about this example, I think, is that long “streaks” of heads or tails don’t seem random to us, even though the probabilities tell us that they can happen.

2) What chance are you thinking of here? I think this refers to the chance that DiMaggio would get at least one hit every 4 at-bats (a typical number for a 9-inning game).

3) This is not quite correct. The Kahneman-Tversky article you are referring to is actually called “The Hot Hand in Basketball.” The point they were trying to make was that people tend to think of “hot” or “cold” streaks (such as long consecutive strings of hitting free throws or 20-foot jump shots) as coming from some particular state in the athlete at the time. Saying you are “in the zone,” for instance, seems to say that you have found some particularly productive mental state to be in, and that is what is causing the better than average outcome. But what the article is arguing is that the same kind of hot or cold streaks would be observed if there were no changes in the athlete, just normal random variation in the outcomes due to chance and probability.

4) Even without bringing in the influence of a higher power, there is the issue of perception mentioned above (see point 3.)