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.