Problem: An asteroid is about to destroy Earth. Humanity has built a spaceship that can fit a fraction of the population. There are two lotteries, you can participate in one only. Both are supposed to have a 1 in 100 chance of you getting a ticket, but you discovered that one was rigged to have a 1 in 50 chance. You want to survive. Which one should you use?
We immediately think that the answer is to apply for a rigged one. When asked to explain, we mention the probability theory and/or common sense. The goal of this note is to discuss why this answer is incorrect and the reasons we are nevertheless inclined to give it.
The key here is that we talk about an existential one-off event (EOE). You don’t get to play this lottery 100 times. And the consequences of the event are massive. The tools and decision-making frameworks of probability theory are built to operate with a series of events, not with a single EOE. Remove the sums and integrals and the formulas fall apart. These frameworks can’t give us guidance on how to act in the case of EOE because they weren’t built for it.
Well, maybe the formulas can’t operate with an EOE, but that doesn’t mean the whole concept of probability is meaningless here, right? It is common sense, after all.
Let’s discuss if it is by thinking through a car insurance problem. A friend asked you personally to write a policy for her grandma’s car. Her standard rate charged by an insurance company is $50/month, and the maximum coverage is $200,000. Your friend will pay you $300/month for a few months. Will you agree?
From the p.o.w. of probability calculations, this is a great deal. The expected payout in any given month is below $50 (the insurance company calculated that for you), and you get to pocket $300. But you kind of feel that this is not a good idea. You may be on the hook for $200k if something bad happens.
So what kind of fee would make you comfortable? First of all, you are likely to politely decline. But if forced to write such a policy, you would ask for a fee of $200k, regardless of grandma’s age, car’s condition, miles driven, and other things affecting the probability of an accident. Your gut tells you that since this is an EOE (you don’t plan to insure millions of cars and $200k is a huge deal for you) the probability concepts don’t really matter/apply to your decision-making process.
To summarize, both, formulas and common sense point to one conclusion. Probabilities are meaningless when dealing with an EOE. (Note: “meaningless” = can’t be used to guide our actions). But why are we then so inclined to use them in the ticket problem? Two notes here.
First, we encounter just a few EOE events during our life, while we deal with millions of non-existential one-offs. And one of the miraculous properties of our reality is that when dealing with multiple non-existential one-offs we can apply the probability calculations to each of them (despite them being one-offs). So we are trained and maybe evolutionary wired to apply probabilities to every event we encounter, repeated or a one-off, by default.
Second, we do have an intuition about EOEs (trained, wired?), as demonstrated by the car insurance problem. We walk away from them. Another example of this intuition at work: a colleague asks you to lend them all your life savings for a year to start a business. No collateral. What kind of interest would you charge? The answer is you would walk away. You wouldn’t bother with probability calculations because they don’t matter. In real life, we walk away from EOEs.
In the case of the ticket problem, we can’t use this natural strategy to walk away, as the problem forces us to participate. Cornered into dealing with the EOE, we grasp at plausible frameworks we know, hoping they will apply here.
The takeaway: you should have worked to avoid the ticket problem. If you must choose, there is no best course of action. You can pick either. You’re at the mercy of luck regardless.