There loosely are three levels of reasoning about the world with respect to one’s actions:
I don’t know what’s going to happen, and I don’t want to; let’s go with the flow.
I understand some basic if-else rules which I could follow to eliminate some paths to shrink my search space.
I am aware of what the probability distribution is, I view the world as some Markov-ish chain of what’s going on, and the way my brain operates can loosely be mapped to traversing the graph of this Markov process.
For a very simple analogy, (1) would be someone who medically qualifies for a moron, (2) would be able to learn tic-tac-toe or chess or Go, (3) would be able to excel at poker or to be an entrepreneur.
Note that what I am talking about here is not the cognitive capacity of an individual, but whether they have internalized certain process of perceiving the world rationally in their mundane activities.
There are people out there who can operate probabilities easily, and yet are unable to do something as straightforward as asking for a raise or changing jobs; we’re not talking about them here. The focus is exclusively on the way people think when it comes to making decisions that directly affect their lives, not some abstract problems they have to solve.
So, what most reasonable people get to is understanding the “highly likely / highly unlikely” events. For example:
This job is way over your head. If you go and interview, it’d be a miracle if you get it.
This person drives a lot while being drunk. It’s most certain that they are going to get in trouble with that. No way I am getting into their car.
Those guys are going out together for a while, and both have their leases expiring soon. There’s a decent chance they are moving in together starting this Fall.
Candidate Trump just talks so much nonsense, there’s no way he could win the elections.
For judgements on life situations like the above, the human brain is generally capable of categorizing them into the “highly likely” / highly unlikely” buckets. If something doesn’t go our way, we get somewhat surprised, but since it does happen from time to time, it doesn’t break our mental model of the world.
The above is how most people appear to live their lives. What mathematicians and statisticians get is that there is a tangible, noticeable difference between something having an 0.01% chance of happening, 0.1% chance of happening, and 1+% chance of happening, and we are able to incorporate these subtle differences into our decision making processes.
For an obvious example, imagine you can have something done in five minutes with a 99% chance success rate, but in that one percent chance it doesn’t work out you’d have to spend half a day cleaning the mess.
If that’s your work you’re doing five days a week for eight hours, and if you’re an average person, there’s a good chance you’ll subconsciously converge to either “I’m doing simple things one after another, and sometimes, oh, damn it, I have to clean the mess, but whatever”, or “My job seems to be all about cleaning the mess, because every single week something goes wrong”.
Someone who understands statistics would immediately realize that if one of a hundred times five minutes become four hours, which is 240 minutes, then of a hundred times there will be 99 * 5 = 495 minutes of cruising and 240 minutes of fixing shit. Thus, with these odds, they’d be destined to spend roughly one third of their business hours cleaning up that shit. Hence, it’s perfectly justified to invest into improving the process, because, for instance, if spending six minutes instead of five and improving the odds of success from 99% to 99.5% would make this workplace significantly more enjoyable, it should absolutely be done.
Basically, people who understand statistics view all or most life opportunities as “X risk Y reward”, and are not hesitant to aggressively optimize the shape of this curve by picking which opportunities to engage in and how to balance them.
For some people the strategy is “as many as possible short-term high-risk high-reward activities”, for some it’s “a few high-risk high-reward, and optimize the rest of my life aggressively so that the worst case scenario is not that bad”. YMMV, but the very idea of consciously and correctly comparing low probabilities to low probabilities and high probabilities to high probabilities is what, in my experience, people who are not proficient in statistics are having troubles doing.
I’m sure the above could be phrased better, but the TL;DR is that for most people the [0.0001% .. 0.1000%] and [99.9000% .. 99.9999%] probability ranges are just points close to zero and one respectively, while for people proficient in statistics et. al. they are extremely wide opportunity windows waiting to be utilized accordingly.