The way you frame a problem influences how you seek to solve it.
I asked ChatGPt, "please give me a simple example of a stochastic [probabilistic] problem as contrasted with a deterministic one," and it offered the following, which I find helpful. What do you think?
Scenario: You have a simple route to work that is exactly 10 miles, and you drive at a constant speed of 30 miles per hour.
Question: How long will it take you to reach work?
Solution (Deterministic):
In this deterministic problem, the outcome (20 minutes) is certain because there is no variability in speed or distance.
Scenario: You still have a 10-mile route to work, but now there are factors like random traffic patterns and weather that affect your travel time. Some days there’s heavy traffic, and on other days, the road is clear.
Question: How long will it take you to reach work, on average?
Solution (Stochastic):
Assume that on average, with light traffic, you can drive at 30 miles per hour, but heavy traffic may reduce your speed to 10 miles per hour.
You can model your travel time as a random variable based on the probability of encountering heavy traffic or light traffic. For instance:
The expected travel time, in this case, would be:
0.7 × 20 + 0.3 × 60 = 14 + 18 = 32 minutes (on average)
In this stochastic problem, the outcome is not fixed and varies due to random traffic conditions. Instead of a single answer, you may need to describe the outcome using probabilities or averages.
The problem is that same, how long will it take, but the answers differ greatly because one assumes simplisity while the other recognizes possible complications. Generally speaking, people don't like complexity. This is why we tend to disregard nuances and focus on what seems obvious to us. In doing so, we may see a complex situation in simplistic terms and so generate deeply flawed solutions.
Because most people prefer simplisity, it's often effective to use complex problem solving to find the best solution and then represent that solution in the simplest possible way. Don't give the formula. In the current example, you could say we will be there in roughly 30 minutes. Or you could just make sure you leave more than enough time and build in some time-killing strategies, the way an plane might cicrcle the airport to compensate for a strong tail wind.