One Run Baseball Luck?

My favorite baseball team has won a lot of close games this year, and that has been a key component in what so far has been an outstanding season. Conventional baseball wisdom, to which I subscribe, has it that a good team does well in games decided by one run. But I wondered what if it is just chance which team wins 4-3 or 9-8? Would some teams win significantly more if the results were just a coin flip instead of skill? I undertook an experiment, which I readily admit is filled with holes, but the results are interesting.

First came in-depth, five-minute Google research to find how frequently games were decided by the slimmest of margins. One source reported that over a decade, 29% of the games were won by one run. Another source for one season had it slightly lower. I was surprised that the percentage was that high. I “confirmed” these rates by looking at the results on one day when all teams played. Four of the fifteen contests were decided by one run, for a rate of 27%. Over the 162-game season, I decided that on average a team was involved in about 46 one-run games.

Then to determine what the results might be in those contests if the outcome was a matter of chance, I decided to flip a coin forty-six times for each of the thirty major league teams. Of course, I was not really going to do this burdensome, boring activity (I might develop a callus on my flipping thumb). Instead, I found an online coin-flipping site that randomly generates heads and tails. This simulated flipper (there are many similar sites) allowed me to choose from a single flip to a string of tosses. Their nearest choice to my desired series of forty-six was fifty, and I decided that was close enough for my purposes.

Making my own random selection, I decided that each heads would represent a victory. When the results were in, I was not surprised that many of the trials clustered around twenty-five heads. More than half of the 30 “team tosses”–sixteen–produced “victories” of twenty-three to twenty-seven games. Perhaps these outcomes might affect a pennant race, but not very often. However, four of the sets of fifty tosses had from twenty to twenty-two heads. By chance, my hypothetical teams would have been six to ten games below .500 (twenty heads—victories means thirty losses, or ten games below .500 in one run games) and that certainly could affect who would win a division. Similarly, six outcomes fell from twenty-eight to thirty heads or six to ten games .500 (thirty heads or victories means twenty losses) in these simulated close games.

Four of the outcomes of these flipping sets of thirty, however, were particularly striking. On the negative side, one team would have won only nineteen games out of fifty by chance and another only fifteen. A manager with a record in one-run games of fifteen wins and thirty-five losses should start looking for another job.

On the positive side, one set had thirty-one heads and the remaing set had thirty-three. Thirty-three and seventeen in one-run games by chance, but this team’s manager would probably win awards and the team might win the pennant.

This exercise reminded me again of how unintuitive probabilities are for most of us. There is a reason why the study of probabilities came late in the modern development of mathematics and basic statistical tools such as significance testing are even more recent. And my thoughts, as they often do, turned to my admiration for the spouse.

The spouse—bless her eager heart thirsting for more knowledge–decided, yet again, that she would try to understand calculus. Wisely, however, she thought that she should first master precalculus. She has been taking online classes, and I have heard much talk about sines and tangents, matrices, and vectors. Now she is in a probability unit. Perhaps our conversation would have amused you as we struggled with what seems to be a simple problem: Using a standard, well-shuffled deck of cards, what are the odds of finding a spade by drawing two cards without replacing the first one back into the deck? If you instinctively knew how to calculate that, you were much better with probabilities than we were.

While that deck-of-card problem was cracked after considerable effort, flipping coins seems clearly straightforward. The chances of getting heads is fifty percent. We know, however, that flipping a coin ten times does not always yield five heads or that flipping a quarter fifty times does not always result in twenty-five heads. However, our instinct is that the result will be close to twenty-five. “Outliers”–such as thirty-three heads and seventeen tails–will be rare. Then there is the issue described to me by a Harvard statistician. He would assign his students to flip a coin a hundred times and record the results. He said he could tell when someone did not do the exercise and just made it up because their lists would seldom have significant runs of heads or tails while real flipping almost always produced, for example, six or eight heads in a row. Therefore, fifty flips would sometimes result in the number of heads well above or below the twenty-five number. Our intuition may be that fifty flips is a large enough trial to eliminate “outliers,” but it is not. And in case you might think that the internet flipper was flawed, out of the total of fifteen hundred tosses–fifty for each of the thirty team–the total number of heads was 748. With fifty flips, however, “outliers” were not really outliers.

However, when I have told my baseball friends about my exercise and say that chance alone might have a team win or lose an inordinate number of one- run games, they rebel at the notion and assert the baseball wisdom that good teams win more one-run games than they lose. It is not simply a matter of luck, and even after my experiment, I still tend to agree with them.

“For Example” is not Proof. Yiddish Proverb (continued)

The anti-vaccination movement not only indicates the power of anti-government feelings, but also important aspects of human thinking in general. We don’t like to live with uncertainties and the unknown. We want answers. My child has autism, and I want to know why. What caused it? For many, “we don’t know” is not acceptable, and they seek some sort of answer. Johann Wolfgang von Goethe said, “Mysteries are not necessarily miracles.” But for bad things, we look beyond miracles and often settle on conspiracies. Autism had to be caused by something, and the vaccine is blamed. We will make up an answer if no answer is available.

Misinformation that has taken root is hard to eradicate. Once we believe something, we want to continue to hang on to the belief. Instances of this are legion, and all of us can cite particulars. This trait permeates all strata of society. For example, a recent article by Gina Kolata,, reports that nearly 400 routine medical “practices were flatly contradicted by studies published in leading journals.” An example: Ginkgo biloba is widely promoted as a memory aid, but a large study published in 2008 “definitively showed the supplement is useless for this purpose. Yet Gingko still pulls in $249 million in sales.” In spite of the scientific research, the gingko biloba myth lives. What T. H. Huxley said about science, indicating how it advances, should be a benchmark for all of society: “The great tragedy of science—the slaying of a beautiful hypothesis by an ugly fact.” But all too often facts do not win out over what we believe.

Moreover, misinformation can easily take root because our instinctive reactions to empirical propositions are often wrong and because our education often does not do enough to train us to think more clearly about empirical propositions. The ground-breaking work of psychologists and economists Daniel Kahneman and Amos Tversky shows that human judgment and decision-making are frequently flawed in predictable ways. Of course, many people know of their studies, but they should be even more widely known than they are. Their research applies not merely to some academic or specialized field, but to all of us as we make judgments about the world in all aspects of our lives. Kahneman and Tversky’s seminal book written with Paul Slovic in 1982, Judgment Under Uncertainty: Heuristics and Biases is not only readable but also fun as is Kahneman’s 2011 Thinking, Fast and Slow. Critical thinking should be part of our basic education, and the wit, wisdom, and research of Kahneman and Tversky should be included.

Critical thinking would also be advanced by the increased teaching of probabilities and statistics in our schools. This thought stems from my own high school math education, which had what might be considered the usual geometry, trigonometry, and algebra. These all helped my thinking, and while I have probably used that geometry more than I realize, I am like many others who say trig and algebra haven’t cropped up much in their adult lives. But I also took a course that had units that included calculus, set theory, and probability and statistics. As I look back, I realize that probability and statistics has been most important to my thinking through the years.

As most of us do, I encounter news of polls and medical and scientific studies, and their meaningfulness requires some understanding of probabilities and statistics as well as does much of the sports information I absorb. But most important was that P & S taught me important things about critical thinking in general, and it taught me to recognize my own and others’ sloppy thinking when we make factual assertions.

Even before reading Kahneman and Tversky, I had realized how bad intuitions, including mine, were about empirical propositions and that something more objective than gut feelings were needed. Probabilities and statistics helped me realize, for example, that a temporal sequence does not necessarily mean a cause and effect and that correlation and causation are separate concepts. Nevertheless, we all too easily fail to recognize our flawed causational judgments. A case in point: Just because autism is diagnosed after the MMR vaccine is given does not establish that the vaccine causes measles.

I learned from probability and statistics that comparisons are often needed to advance empirical knowledge but devising and assembling adequate control groups can be a tricky business. And we need to have some understanding of the statistics used to make the comparisons.

However, many of us who have not been deeply trained in science and math simply throw up our hands when we encounter either and fall back on some shortcut to determine what we decide to believe. My law school teaching showed me that time and again.

(concluded July 26)