Playing the Percentages

Author's note: I am not a medical doctor. If you have a specific question about the diagnosis, treatment, or prevention of any illness, please talk to a medical doctor. That's not just an attempt to shield myself from liability; it's a genuine statement of my views on how we delegate responsibilities in society. I wouldn't want anyone taking medical advice from me any more than I'd want them taking medical advice from some quack with a YouTube channel. This article is presented not as a medical article, but as an article on risk management. And baseball. 

A summer afternoon in the ballpark has a way of engaging the senses. The grass is an impossibly deep green against the muted red of the infield dirt. The home team uniforms are crisp white, accented with the team colors. The road team's grays are similarly accented. The warm air carries the scent of steaming hot dogs. Fans shell peanuts and wash them down with ice-cold cups of foamy beer. 

There are no outs and nobody on; the hitter has a count of one ball and two strikes. The scouting report says this hitter makes a disproportionate number of his outs on ground balls. The catcher puts down three fingers. The pitcher goes into his windup—carefully practiced to look exactly the same no matter what pitch he throws—and brings the ball past his ear, gripping it on the side and imparting a spin whose axis is parallel to the ball's direction of travel, the way an American footballer would throw a spiral. The batter, who has only a tenth of a second to decide whether to swing, sees what looks indistinguishable from a fastball leave the pitcher's hand, and senses his opportunity. Just as the batter swings, the baseball, in response to unseen aerodynamic forces, veers suddenly downward. The batter tries to adjust, but it's too late. The lower portion of the bat strikes the upper portion of the ball. The ball moves sharply across the infield, but everyone in the stadium already knows that the pitcher has won this particular contest-within-a-contest. He got the hitter to chase the slider, and induced the kind of ground ball that he knew going into it would result in an out. The shortstop—always the best athlete on any team—makes a couple steps to his left and softly absorbs the ball into his glove. He knows it takes four and a half seconds for the batter to run from the batter's box to first base, and he adjusts on every play to use all of those four and a half seconds. No sense rushing if you don't have to. In one motion he smoothly transfers the ball from his glove to his throwing hand, makes certain that both feet are planted firmly in the muted red infield dirt, and throws the ball to the first baseman. The ball hits the first baseman's glove when the batter is a step and a half from reaching the bag.

If you were watching closely, you'd notice something else. As soon as the bat made contact with the ball, the catcher came out of his crouch, removed his mask, and ran alongside the first base line, almost parallel to the batter's path, to place himself about fifteen feet beyond the point where the first baseman caught the shortstop's throw. He's there to back up the first baseman. If the shortstop's throw goes wide, the catcher should be able to get it in his glove and prevent the batter from advancing to second.

What's happened here is obvious. We just got done describing the shortstop as the best athlete on the team. Well, is he or isn't he? Either the shortstop is capable of making an accurate throw to first or he isn't. If we need the catcher—who historically has been the smartest player on the team but far from the best athlete—to handle his errant throws, then the shortstop must be worthless. We should eliminate the position. Just field eight players. Actually, the same logic could be applied to the rest of the fielders as well. They're worthless, all of them. Get rid of them all. Put a pitcher on the mound, and a batter in the box, and forget about all those other players who we all know aren't really contributing. We could earnestly play three true outcomes baseball, and the owners would save a fortune in player salaries.

The previous paragraph is deliberate in its absurdity. Playing shortstop is not an all-or-nothing proposition. Shortstops make accurate throws to first with uncanny reliability, but not to perfection. The best shortstop in the Majors will have a fielding percentage well above .990; the worst will still be above .950. Over a large enough sample, no shortstop would be expected to have a fielding percentage of 1.000; no human activity ever takes place without the risk of failure. The shortstop is not perfect, but far from worthless, and that tiny imperfection is where the catcher comes in. Let's look at the probability that what should be a sure ground out will instead be a runner in scoring position. To be sure, the most desired outcome would be to record the out at first, but for now we'll study the question of whether the team can prevent the runner from reaching second, whether by throwing him out at first, throwing him out at second, or discouraging him from trying to advance to second after reaching first. Suppose the shortstop has a probability of overthrowing first base of 0.003. Allowing for the encumbrance of his gear and the fact that an errant throw could go just about anywhere, let's give the catcher a probability of failing to recover that throw of 0.050. Without the catcher backing up the first baseman, there's a 0.003 probability of having a runner in scoring position; with the catcher in place, we multiply the probabilities of each player missing the ball to get an overall probability of failure of 0.00015. A small risk has been converted to a trivial risk. Since long before sabermetrics ruined—I mean radically altered—the game of baseball, managers have known these mathematical realities. They have an expression for the small things a team does to improve its probability of winning; it's called playing the percentages. 

Playing the percentages isn't limited to baseball. Aside from Benjamin Franklin's observation about death and taxes₁, there pretty much are no sure things in life. In every field of human activity, we find ourselves faced with uncertainty, and we come up with ways to make those uncertainties manageable.

In engineering, the term for chaining together opportunities for failure in such a way that overall failure is contingent on failure at each opportunity is called redundancy. Redundancy takes advantage of the same math we saw with the catcher backing up the first baseman. Each opportunity for failure has a probability of failure between zero and one; multiplying those probabilities together results in an overall probability of failure that is much smaller than that for any one of the opportunities. As you chain together more opportunities, the overall risk of failure approaches zero. This is why automobiles have seat belts and air bags. It's why airliners consume radar services from air traffic control and also carry their own traffic collision avoidance system (TCAS). Nobody would assert that the existence of TCAS is a tacit admission that air traffic control is worthless; airplanes are magnitudes of order safer when they use both.

As of this writing, just over 60% of the United States population is fully vaccinated against COVID-19. In the polarized, post-objective-reality world we find ourselves in, it's actually somewhat surprising that we can get 60% of the population to agree on anything, let alone take the specific action of getting an injection that up until a year ago didn't even exist. Vaccination rates vary widely by state, though, and the narrow consensus that we have been able to hammer out is of little consolation to overworked health care workers in regions of severe outbreaks, or to those with serious illness and their families. Vaccine hesitancy in this country remains frustratingly persistent.

The reasons for vaccine hesitancy are all over the map, from the somewhat reasonable (the speed with which this particular vaccine was brought to market causes some to worry about the thoroughness of the safety and efficacy studies) to the outlandish (Bill Gates is planting nanochips in the vaccine to track your movements and control your mind). One that has gained prominence lately takes the form of my argument for removing the shortstop from a baseball team. Those of us who are vaccinated are encouraging the rest of the population to also get vaccinated. In this particular vaccine hesitancy story, that is seen as a tacit admission of the vaccine's inefficacy. Either a vaccinated person is protected or not, and if you're protected, then you shouldn't have to be concerned about contact with those who are unprotected. It doesn't work that way, and the reason why it doesn't work that way is the same reason as why the catcher backs up the first baseman.

Consider two fans at our hypothetical baseball game: Bill and Ted. Bill and Ted don't know each other; they just happen to have adjacent seats. Bill is fully vaccinated. Bill wants to know what the probability is that Ted will transmit the virus to Bill. Working out the math behind how a virus goes from one person to another is complex, and I already said that I'm not an MD. For simplicity, let's isolate three factors to calculate the probability of transmission. First, there's the probability that the vaccine will fail to protect Bill in this instance. That number is close to zero, but not identical to zero. We'll call it P_f. There's the probability that Ted is infected, which would be between zero and one. We'll call it P_i. Then we'll use a third probability to stand in as the product of all the factors that I'm not qualified to accurately measure—or even describe. We should all be able to agree that that is also a number between zero and one; we'll call it P_e. The probability that Bill will get the bug (we'll call it P_t) is the product of the three:

P_t = P_f * P_i * P_e

If Ted is vaccinated, P_i is much lower than it would be if Ted was unvaccinated, and consequently P_t is much lower. Bill is well protected with the vaccine (P_f is almost zero with the vaccine, and is one without it), but he is orders of magnitude better protected if Ted is also vaccinated. Ted might not care about Ted (or Ted might say he doesn't care about Ted), but Ted might take a moment to consider the real person that Bill might be. Maybe Bill has an elderly grandparent. Maybe Bill has an immunocompromised kid. Maybe Bill is a health care worker, and being down with an illness—even if it's a mild case—would further complicate an already complicated situation. Maybe none of these describe Bill, but there's an excellent chance that he's only a few degrees of separation away from someone who does. 

Vaccines work, but they're not an all-or-nothing proposition. Because of that, we can all make a contribution by getting vaccinated. Stay safe, be well, and have a wonderful Holiday with your family or whoever you hold dear this time of year. On an almost unrelated note, let's see some baseball next year.

1. A coworker once pointed out that death is dependent on life, and taxes are dependent on gainful employment. The only two things that are certain, he concludes, are life and gainful employment.