After the outbreak in China of the COVID-19 virus, Italy has fallen in the middle of the most acute sanitary emergency we ever experienced in a long while. And what's worse, other countries are sadly joining it. As I write this piece, over 15,000 Italians have tested positive to the virus, and over 1000 have already died. Based on very convincing analysis of data from China, we know that the real number of cases is higher by at least a factor 20, if not 100. In these conditions, individuals have to protect themselves and help reduce the spread of the virus with all possible means - most importantly, by staying home and cutting all social contacts.

Despite the emergency, it seems the global response to this pandemia is very slow. Most individuals do not realize the danger and pretend to continue living in the same way they did before; they feel that preventive measures indicate paranoia rather than reasonable behaviour. In this post, I will try to explain why being "paranoid", as some would claim, is actually the only reasonable thing to do. The point is a very simple one to make, but must be elucidated in full: SMALL PROBABILITIES ADD.

Let us say your friend Joe is afraid to fly. You probably consider his reaction to the idea of being airborne as irrational, and if you do you are right: the chance to die if you take a commercial flight these days is smaller than one in a million. A millionth is a very small number, and Joe is being irrational in not taking that very small risk, as there are many other risks he does not avoid taking which are comparable or larger. In Bayesian terms one would need to factor in the loss Joe faces if he does not travel, in order to understand what is the best choice for him; but in general, the benefit of taking the flight versus not taking it is much larger than the small probability one faces of getting killed. Hence air travel is popular and rightly so.

But now let's say for the sake of argument that you have to decide whether to fly four times a day for the next year (and let's forget about the stress you would accumulate if you did). When deciding for or against that plan, would you consider the risk that one of those flights ends up in tragedy? The point is that while each of the flights has a one-in-a-million chance of doing that, the global probability that at least one of them does is much larger: if we are talking about 1460 flights, we must combine these numbers. And here Statistics comes in.

A simple calculation

Since the outcome of every flight can be considered an independent event from all others, we may combine probabilities by a simple calculation. If P is the probability of one event, the chance that the event does not take place is 1-P. To survive two flights you therefore have to multiply (1-P) for the first flight by (1-P) for the second. To survive 1460 you have to multiply (1-P) 1460 times. What you get, in case P is one millionth, is 0.998541: so there is a probability that at least one of those flights kills you, globally, of one minus 0.998541, so 0.001459: a one point five in a thousand chance. And here you may notice two things.

The first is numerical: since P was very small, the operation of elevating to the 1460th power (1-P), and subtracting the result from 1, is numerically almost equal to computing 1460 times P (1460*0.000001=0.001460): in other words, small probabilities add, to very good approximation. If you are concerned about something rare happening to you, which has a small chance of occurring if you perform some action, you may estimate the chance that it does happen, at least once, by adding the small probability as many times as you perform it.

The second point is that a 0.15% probability of dying a sorry death is not negligible. At least, I would take it seriously. If you are like me, you would probably think twice before taking the risk.

Now, let us take the case of the pandemic of coronavirus. You are at home, the fridge is empty, and you have to get groceries. You know that out there there are contagious people; but you have been told that it is okay if you stay at a distance, wash your hands frequently, etcetera. Still, there are small probabilities that random events cause you to get infected. And once you do, your chance of dying of pulmunar complications is in the 2-3% range. Not negligible! Yet if you protect yourself chances are you will not get infected. But let us now discuss each and every action you take, and see if we would call paranoid a very careful behaviour.

For sure, you should wear glasses and a mask. These will reduce the risk if you go out there, although they will not provide full protection. The risk is that, even keeping at a distance from everybody else, you walk into a space where somebody coughed a minute before: the droplets of water, filled with virus, are still lingering there in the air in the form of an aerosol, and you may inhale them or get them into your body through your eyes. Mask and glasses will reduce that risk, although they will not eliminate it. Do you consider somebody walking with a mask as a paranoid person? I bet by now you started to change your mind with respect to that evaluation.

But should you grab that door knob with your unprotected hand in order to enter the shop? The virus could have been deposited there by a previous customer. Should you wear gloves then, or do you call that paranoid behaviour? For sure, if you do not use a protection, you have a very small chance that your hands get contaminated. Once you get back home, provided you did not touch your mouth or nose in the meantime, you will wash them, but maybe a chance still exists that you will still get infected. A small one, to be sure. Should we protect against it with gloves?

Then, as you are walking back home, a person is coming in the opposite direction. The walkway is six feet wide. Do you change your direction to avoid crossing that guy? The chance that he is infected with the virus is of the order of one in 100 if you live in Italy now, or smaller if you are elsewhere. The chance that he coughs as you cross is even smaller. Do you call paranoid the behaviour of somebody who makes a detour to avoid the possibility?

I could go on, but as you certainly understand, by interacting with an infected environment we are subjected to thousands of very small events, each of which has the potential to get the virus in our body. We might call paranoid the behaviour of somebody who singularly addresses each and every one of those occurrences, and we might be right on each individual event, but we have to look at the global effect of small probabilities. They add. So, if your chance of getting infected by touching surfaces while not wearing gloves is one in 10 thousand, and you touch 100 surfaces, your odds of getting infected become one in a hundred. If you meet 50 people in the street, and each of them could be infected and cough, with a one in ten thousand chance of transmitting the disease to you, the total odds that you get infected are half a percent.

We are currently in a situation where our individual behaviour will determine whether in six months we will have 100 thousand, or a million, or 50 million deaths by the COVID-19 pandemia. So please, ignore the awkwardness of looking paranoid: act paranoid and save yourself and others. Stay home, unless absolutely necessary. And think of every possible risk, taking it seriously however small. Because, you now know better, probabilities add.

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I would like to end this rather alarmistic post with a positive note: China, which fought the virus by applying extremely rigorous measures, and has now apparently managed to contain it, sent to Italy a team of experts and tons of sanitary goods to help us fight the emergency. They did it unprompted, and for free. I highly commend these brave physicians and the solidarity of China in a difficult moment for Italy and for the world. Below is a picture of these 21st century heroes.