If you have a sense of what the future could look like, you can make decisions today to bend tomorrow towards benevolence. When it comes to investing, government planning, research funding, personal life, and more, we make decisions based on hard and fast rules, stochastic processes, and the unknown. While it is possible to reason about how to deal with the laws of physics and make predictions about probabilistic phenomena, the playbook for handling uncertainty requires a bit more intuition. When it comes to immeasurable unknowns, what is probable needs to be redefined and reconsidered as what is possible. This is especially true when you suspect an unknown is driven by exponential growth.

Starting with a simple hypothetical casino game, this essay explores how an exponential distribution can lurk behind what appears to be a profitable looking gamble. What looks like a good series of bets in theory can turn out to be quite dangerous in the real world. The difference between additive and multiplicative growth rates changes the nature of decision making. Multiplicative growth rates can create probability spaces that grow faster than they can be explored in time, which makes a system prone to ruin. In the blinding light of uncertainty, the precautionary principle—a refocusing of cost-benefit calculus to survival preparation—is a necessary and courageous heuristic to apply to a class of rare phenomena with deadly consequences that tend to be driven by hidden exponentials. Some of these phenomena include epidemics, terrorism, and climate change. We will consider the ongoing coronavirus pandemic and why we should treat novel viruses (especially fast spreading ones that can kill people) very differently than the flu. When we identify potential hidden exponentials, we can make choices that limit our downside. Sometimes, we can even make choices that offer unlimited upside.

## In the real world, expected value can be less than expected

How do you make decisions in an uncertain world? It depends. If you can quantify your uncertainty, then decision making boils down to weighing tradeoffs between benefits and costs. This pragmatic tradeoff considering all potential outcomes and their likelihoods can be represented as expected value under the law of large numbers.

Now, let’s imagine that we arrive at a casino on its opening night, and we sit down to play a simple game based on flipping a coin. The rules are simple. You can bet whatever you want. If the coin comes up heads, then you get your bet back plus another 50%. If the coin comes up tails, then you get back only 60% of your bet.

Would you want to play this game? Given that heads and tails are equally likely outcomes, you gain more when the coin comes up heads than you lose when the coin comes up tails, and you can bet whatever you want; this could be a really profitable game. We can project how profitable a single bet will be by calculating its expected value, which is the mean of the probability distribution. Here, it is 105% of the bet. In simple mathematical notation, the game can be represented with probability distribution F(x) and expected value E(x).