People misuse and misunderstand probability constantly. The mathematical meaning of the word is different from how most people use it in conversation. When someone says “he’s probably thinking about food,” they don’t mean probability in any formal sense. They’re speculating based on what they know about him.
A better example, and a more common misconception, is the classic game show setup. One door has a car behind it. The other has nothing. Say the car is behind door 1. The incorrect phrasing people use is “there’s a 50% probability the car is behind door 1.” That’s wrong. There’s a 100% probability the car is behind door 1 and a 0% probability it’s behind door 2. Actually, it’s not a probability at all. The car is certainly behind door 1. When you’re dealing with something that already is, you aren’t talking about probability. Probability refers to future events that might happen.
So when someone says “there’s a 50% probability the car is behind door 1,” what they actually mean is “there’s a 50% probability you will pick the door with the car.” That’s correct, because it describes a future choice. But once you pick a door—say door 1—the so-called probability changes. You’ve now picked the door with the car with 100% certainty. Did the probability change? No. You were just using the wrong word.
Here’s an example where the language works correctly. Take a jar with 15 balls: 5 red, 5 green, 5 blue. You can correctly say the probability of picking a red ball is 1/3, or roughly 33%. This is a legitimate use of the word, because it refers to a future event—the act of reaching in and pulling one out.
This loose misuse of probability leads to errors even among people who should know better. Conspiracy theories are a good example. Most turn out to be wrong, but some occasionally are true. Say, hypothetically, that 5% of conspiracy theories throughout history have been true. That’s a made-up number, by the way—I have no idea what the real figure is. Now say a new conspiracy theory surfaces. The temptation is to say “there’s a 5% chance this new theory is true.” That’s wrong. The theory is either true or false right now. It doesn’t have a probability. What people actually mean is that conspiracy theories, as a category, have a track record of usually being false, with only a small percentage turning out to be true. That’s a statement about history, not about probability.
So why do people use the language of probability anyway? They like to reify things—to attach a number to something that feels otherwise unmeasurable. It also helps them feel justified. The person who enjoys debunking conspiracy theories loves to say “statistically, 95% of conspiracy theories turn out to be false, so we should reject this new one, since there’s only a 5% chance it’s true.” As we’ve seen, that doesn’t hold up. Each claim should be assessed on its own merits and the evidence available, not on the historical batting average of an entire category.
Here’s a more personal example. Suppose you have some concerning symptoms and start googling them. Your symptoms match a few things—most aren’t serious, but one is serious and rare. You look up how rare, and find that the incidence is 5,000 out of 100,000 people, or 5%. You might reason that there’s a 95% chance you don’t have it and feel better about the whole situation. But are you actually justified in feeling relaxed?
Here’s where we need to be careful. Statistically, since only 5% of people ever get this condition, it’s unlikely you will get it. Notice the word “will”—that’s a statement about the future. But right now, you either have the condition or you don’t. That’s always true, whether or not you have symptoms. This is why doctors rule out serious causes first, no matter how rare, and why they recommend checkups even when you feel fine. It’s also why you shouldn’t rely on a misapplied version of “probability” to figure out whether you have a serious problem.
But back to the question: are you justified in using probability to reassure yourself, something like “I probably don’t have that. It’s so unlikely. There’s a 95% chance I don’t!” It is rare, so the odds of any given person getting it are low, and you are, in fact, some given person. It feels intuitively right to tell yourself you probably don’t have something that most people never get. But is it actually right? I think it’s almost right, but it has nothing to do with probability.
There are countless bad things that could happen to you. A serious disease is just one of them. There’s also lightning, car accidents, crime, dangerous insects, foodborne illness, and so on. You don’t sit around running probability calculations for all of these. It’s only when a specific concern grabs your attention that you start reasoning your way out of the anxiety it causes. Misusing probability this way is a coping mechanism, not an assessment. You aren’t worried about the dozens of other bad things that could happen to you, so why worry about this one? It’s not really about the numbers. It’s about what you’re focused on.
Instead of telling yourself “I probably don’t have this,” which leaves the door open for anxiety to creep back in, try “I don’t know, but I’m going to find out.” This removes the false comfort of probability from the equation entirely and frees you to look at actual causes. What are the risk factors? What do people who have this condition have in common? Leaning on probability here is lazy. It gives you a false sense of security or a false sense of fear, depending on which numbers you land on. Probability applies to unknown future events. Once something already is, and once you know it is, probability becomes irrelevant.
Here’s a cleaner way to see it. If you flip a coin without looking at it, then guess heads or tails, you have a 50% chance of being right. What you can’t say is that there’s a 50% chance the coin landed heads up. The coin has already landed. That chance is gone. Probability only applies while you still have a future opportunity to guess. Once you make your guess—say tails—you no longer have a 50% chance of being right. If the coin landed tails up, you’re right. Not probably right. If it didn’t, you’re wrong. Not probably wrong.
To make this more intuitive, think about something that already happened. Suppose yesterday’s forecast gave an 80% chance of rain, and it rained that evening. If someone told you today, “there’s an 80% chance it rained yesterday,” you’d immediately recognize that as nonsense. It rained. Chance and probability no longer apply.
But don’t we routinely prepare for things that seem statistically more likely, even without any sign of an actual problem? Sometimes, but only when missing something could be catastrophic. Doctors check your heart even when you have no symptoms of a heart attack. They don’t check for a splinter between your toes. Splinters are far more common than heart attacks, but missing a splinter doesn’t matter. Missing a heart problem does. When the stakes are high, probability isn’t the filter you should use to dismiss the question outright.
Featured image by Ben Mathis Seibel on Unsplash