Gun rights advocates are correct: a well armed principal might have reduced the death toll from the tragic elementary school shootings in Connecticut last week. Gun carrying citizens might also have been able to take down the shooters in Aurora and Virginia Tech. To most people, after all, guns are about self-defense, not about committing crimes. As the old saying goes: “There has never been a mass shooting at a gun show.”

On the other hand, gun control advocates are correct to point out that mentally disturbed people like Adam Lanza would not be able to commit massacres if they were prevented from getting their hands on high-powered, semiautomatic weapons. They are also correct to point out that Americans have staggeringly easy access to weapons that far exceed what any sportsmanlike hunter would use during deer season.

In other words, figuring out what to do in the wake of the Connecticut massacre means recognizing the truth in both of these views. It means considering the possibility that the answer to reducing gun violence is a matter of both having more guns and less. To understand what I mean by “both more and less,” I offer two analogies: a straightforward one about airport security, and a more unexpected one about breast cancer screening.

For several decades now, experts have been working to reduce the chances that airline passengers will carry weapons on board airplanes. This strategy has been extremely successful in reducing airplane violence. But these measures have also been extreme – x-rays and metal detectors; long lines and pat-downs. But airport security does not stop with gun control efforts alone. After all, evil people do not need guns to commit violent acts. Organized martial arts experts could overpower airplane personnel with nothing but their limbs. So we have done something else: we have armed pilots and federal marshals with guns while they fly in airplanes. We have created policies that simultaneously decrease and increase the number of guns in airplanes, with a heavy emphasis on restricting guns, out of a recognition that zero guns is not a workable solution.

Of course airports and airplanes are uniquely vulnerable targets. We tolerate extreme gun-control measures in airports because we have become all too familiar with the consequences of allowing evil people to board planes with guns and even box cutters. We take some of the same measures in some schools, with metal detectors being an increasingly common sight on public school grounds. But there are limits to what we should do to restrict guns in our schools. I doubt any of us want TSA agents to replicate their airport security procedures outside of our nation’s elementary schools. What’s more, massacres have not been limited to school sites, but have also occurred in malls, post offices, movie theaters and college campuses. And finally, keep in mind that we have more guns in this country than we have people. We cannot expect even the most extreme gun-control laws to reduce the chance of massacres anytime soon.

Is more guns the solution to this problem?

The idea of arming US citizens with guns to prevent senseless massacres makes sense when we think of individual tragedies:

• A well armed principal might have stopped Adam Lanza from killing all those children
• A gun carrying movie theater manager might have been able to take out James Holmes

But this idea falls apart when we think beyond identifiable massacres, and consider the broader implications of a more heavily armed populace. And that is where breast cancer screening offers us guidance.

On The Diane Rehm Show on December 12, several experts debated the pros and cons of breast cancer screening. One of these experts was Dr. Wendie Berg, professor of radiology at the University of Pittsburgh. (In the spirit of full disclosure: I spent two years at Pitt in the early 90s and loved the place.) Berg urged women to get mammograms, with emotionally provocative language: “Women… may not see their child graduate from high school if they are diagnosed with advanced breast cancer.”

In addition to this emotional plea, Berg also appealed to a mathematical argument: “Out of, on average, 11 women who are diagnosed with breast cancer, one of these will have their life saved because of the mammogram.”

Let us accept these numbers for the purpose of discussion. Because even if these numbers are factually correct, they are the wrong numbers to think about when deciding whether mammography is a good idea. The numbers are flawed because they focus only on the population of women diagnosed with breast cancer. By contrast, screening mammograms are given to a much larger population of women, in hopes of identifying those who have silent breast cancer. It is impossible to understand the pros and cons of mammograms without looking at the entire population of women who receive this test.

Gil Welch did just this during this same discussion. Welch is a physician at Dartmouth and an expert on the risks and benefits of cancer screening. (He is an incredibly smart dude and, keeping up with this whole disclosure thing, is someone I’ve known for a couple decades now.) Welch acknowledges that mammograms save lives. Out of 1000 women screened annually for 10 years, approximately 1 will have a life saved because of her mammogram. But what about those other 999 women?

• A few hundred will be subject to at least one false alarm
  • Leading to anxiety and painful biopsies
• As many as 10 will receive aggressive treatment for what would have been a harmless cancer

Too often after gun massacres, we think like that Pittsburgh radiologist. We think about the specific identifiable tragedy, the place where we know something bad happened, and we imagine how it could have been a different place if an honest citizen had been armed with defensive weaponry: “If we had had more guns in the hands of responsible professors, the Virginia Tech massacre might not have been so deadly.”

That is the wrong way to think. We need to reason more like Welch – and consider the population as a whole, not just the population unfortunate enough to be in the wrong place at the wrong time. Guns can and do prevent deaths, when used in defense of situations. But more often, guns kill in other ways – through accidents for example. Many more children are accidentally killed by guns they find in the home than by any homicidal maniac.

The same logic is crucial in evaluating well-intentioned policies, such as laws requiring that children be placed in a car seat while flying on airplanes. No doubt, if an airplane has a runway crash, toddlers strapped to car seats will be more likely to survive than those wriggling around on their mother’s laps. But we need to take a broader look before deciding whether this kind of policy would actually save lives. We can’t just think about the population of children, the very rare ones, who are on airplanes that experience runway disasters. We need to think about the entire population of traveling children, many of whom will now be strapped into car seats in their family automobiles because their parents can’t afford to buy them their own seat on an airplane. Highways are much more dangerous, per mile, than airplanes. If we forced parents to buy airplane seats for their toddlers, to make them safer on the planes, we would be killing children, because many of them would be rerouted to the highways with their families, where they are much more likely to die.

In the face of unfathomable tragedies like the Connecticut shooting, it is natural to jump to quick conclusions. That is why gun rights activists quickly imagine a world where well armed teachers defy such assaults.

Thinking that way would be a tragic mistake.

Peter Ubel is a physician and behavioral scientist who blogs at his self-titled site, Peter Ubel and can be reached on Twitter @PeterUbel.  He is the author of Critical Decisions: How You and Your Doctor Can Make the Right Medical Choices Together.