Let’s consider someone named Theo. He studied physics in college, is particularly outspoken against unregulated capitalism, and is critical of the U.S’ lack of universal healthcare. Which of the following is more probable: 1) Theo runs his own business, or 2) Theo runs his own business and he is a socialist? Given Theo’s opinions, we might think that option two is the most probable. It might be surprising, but given what we know about Theo, mathematics points us in the opposite direction. The less specific option is guaranteed to be either more or equally probable to the chance of one and two both being true due to the probability of conjunction. In short, when we need to think about whether we should believe the truth of various claims, and we have no ability to access empirical evidence for ourselves, we should always go with the least specific option until there is overwhelming evidence to believe otherwise.
The “conjunction effect” is when our instincts drive us to make a probability mistake about how likely something is to be true. When we are given information about a situation in conjunction (i.e., A is true and B is true) compared to just one piece of information (i.e., A is true), we are much more likely to believe the more specific option when it fits squarely with our expectations of the world. But as we know from math, the more specific something is, the less probable it is to be true. That is because, to say both A and B are true, we need both independent claims to be true at once. On the other hand, the less specific option is more probable because only one claim has to be true. To emphasize this idea, let’s take a somewhat absurd case. Is it more probable that 1) my car is having engine troubles, or 2) my car is having engine troubles and my house is on fire? It’s certainly more probable that option one is true, because the second option would need something unrelated to my car to also be true.
In math, this rule about probability can be written as Pr(????∧????)≤Pr(????), which merely means that the probability of A and B both being true is less than or equal to the probability of just A being true. My motivation for talking about this stems from a number of troubling conspiracy theories I keep hearing about COVID-19. One that I want to take on here is that “COVID-19 is not real and this lockdown is a plot of elitist ‘deep state’ officials, like Dr. Fauci, to strip away our individual freedoms”. To many of us this sounds absurd right off the bat, but fear, mistrust, and lack of information can have a huge impact on our judgement, regardless of who you are.
Luckily, we can use basic principles in math to help us shoot down wild conspiracy theories that do more harm than good. Let’s break down the above conspiracy. Claim A is “COVID-19 is not real” and claim B is “this lockdown is a plot to strip away individual freedoms”. How likely is it that COVID-19 isn’t real? The odds of that being true approach zero. We have extraordinary evidence of people catching COVID-19; we don’t even need to believe a specific case/death count is accurate (i.e., current numbers could be an under-count/over-count). The moment one person even catches the virus, A is false. Given that the COVID-19 count on April 21st was 2,501,156 people worldwide, it’s safe to say that there is at least one real case–therefore, claim A is incredibly improbable.
What about claim B? This is even less probable than claim A. For B to be true, you would need to have 50 governors, 1.1 million doctors, 2.86 million nurses, and hundreds of thousands of profit-driven, private sector companies pushing the same narrative that we need to stay home for a few weeks (i.e., they are “in on” the plot). All of those people have something to lose: re-election, profit, or their lives. Now, if we put claims A and B together (i.e., there is a plot and COVID-19 isn’t real), out of mathematical necessity, this is at least as unlikely to be true as A on its own or B on its own. Since both claims have a near zero chance of being true, if we put them together the chance is even closer to zero.
Fear makes us latch onto narratives that are more comforting to hear when we can’t control natural events. We are compelled now more than ever to unpack the wild claims that spread across the internet and within our circles of friends. Stay healthy.
Patrick F. Bloniasz is a policy researcher and a Mathematics and Neuroscience student at Bowdoin College.
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