The Mathematics of Making Mistakes

During the past week I’ve been thinking about mistakes. I believe that this thought process originated from my lingering irritation with myself for an error I made in a column several issues ago (for those who missed it, I used the word “median” and the word “average” as though they were interchangeable). It gained momentum with the realization that I am almost never right in the eyes of my teenage son, Andrew. And this little essay finally took shape when I spent some time reflecting on my father.

We are all familiar with the phrase, “to err is human.” Of all the clichés that our language contains, this might be the single most recognized and the most undeniably true. When we are young and make mistakes, our parents console us with the altruism that “everyone makes mistakes,” and usually add that the important aspect of any error is to “learn from our mistakes.” This, of course, is wise advice, which explains why virtually every parent shares this knowledge with his or her children at some point.

As we grow a little older, this advice loses some of the solace it once provided. Mistakes that we make as we mature, particularly about things that really matter to us, become more and more upsetting. Though we can still learn from our mistakes, the errors seem to weigh more heavily on our minds and linger longer in our memories.

The fact that mistakes are made by any of one us should come as no surprise. Far more often than not, life requires us to make decisions before we can possibly have all the information we would like to have, and before we can be certain of all the possible outcomes of our decisions.

In a book by George Schaffner, entitled The Mathematics of Life, he uses the example of emergency room doctors. In life threatening situations, doctors are faced with a myriad of decisions that all must be made in an extremely short time frame. In order to make the best possible decisions they rely on their training, on their experience, and whatever tests time allows and the situation seems to require. More often than not, however, they must make decisions before they have all the information they might like to have…but they have to act.

Our day-to-day decisions are seldom as critical as those of emergency room doctors. Still, we are all faced with a multitude of decisions everyday. Let’s say that we are required to make 10 decisions each day where, either in the near or the long term, we can be right or we can be wrong. That means that we must make 70 such decisions each week, or 3,640 such decisions every year. But this also means that we are faced with a great deal of opportunity to be wrong.

The problem that I have only recently discovered is that, despite our best intentions and no matter how carefully we weigh our decisions, we all make mistakes more frequently than we would like to think. And before you begin to rationalize your personal exclusion from this assertion, or start to generally refute this theorem, I’m going to prove it.

First we need a little background on probability. If you roll a single die the chances that you will roll a three are one in six, because there are six numbers on each die and each number has an equal possibility of appearing (assuming the die isn’t weighted in some manner). The probability changes substantially when it comes to rolling two consecutive threes. In that case the probability is one in six times one in six, or one chance in 36 tries.

Now let’s shift our attention to our day-to-day, right or wrong, decision-making. Let’s say that you are an extremely intelligent, methodical, rational human being (as all my readers are). And let’s say, that because of your superior intellect, you make the right decision in your day-to-day life 80 percent of the time. Most of us, of course, assume that being right 80 percent of the time means that the right decision is made eight times before a wrong decision is made. Unfortunately, this isn’t the case.

If you make the right decision 80 percent of the time, by the time you make your third decision you have only a 51.2 percent chance of being right (0.8 x 0.8 x 0.8). And by the time you reach your fourth decision you have a better chance of being wrong than right (0.8 x 0.8 x 0.8 x 0.8 = 0.41)!

The one factor that probability can’t take into account, of course, is how informed we are when we make our decisions. As long as time will afford us the opportunity, the thorough research and careful consideration of options and outcomes can impact the mathematical certainty. Still, the fact remains that, ultimately, we will make mistakes more frequently than any of us would like to think.

When I was young, I was certain that my father was never wrong. When I would mention this he would say something like, “Actually, I’m only right about 95 percent of the time. But I’m working hard for 96 percent.” As I grew older my idealistic opinion of my father tempered to some degree, but I was still certain that he was correct more often than anyone I knew. His response to my praise altered, as well. “I make mistakes,” he would say, “but never twice in a row.”

If you use the same math that we used above, and assume that my father had improved to a 96 percent correct level, then the likelihood of his ever making two mistakes in a row had shrunk to 1.6 percent. So maybe he was right…once again.

My current struggles with Andrew have much more to do with psychology than mathematics, and he may never hold my intelligence and decision-making ability in as high as regard as I did my father’s abilities, and that is just fine. Still, there are days when I miss his youthful, unwavering certainty that I was always correct.