# Probability — analytical results instead of simulations

After students learn how to use probabilistic simulations, what strategies can one use to encourage them to understand analytical results anyway? For example, I'm struggling to find a compelling reason/example to teach about the maths behind expected values when they can just run a simple simulation. I'm talking about students interested in applying probability, not students who appreciate pure maths.

• how about examples like the Cauchy distribution where the expected value doesn't exist? That has real-world applications. Less realistically, there are easy-to-understand examples where simulated calculations will virtually always be badly wrong (e.g. a random variable that takes the value $10^{1000000000}$ with probability $10^{-100000}$ and is otherwise 1) which highlight possible sampling problems. Commented Sep 27, 2021 at 14:45
• "I'm struggling to find a compelling reason/example to teach about the maths behind expected values when they can just run a simple simulation." If it's obvious that the simulation will very closely approximate the analytical result, then often times there is no point. Also, what about teaching them the maths behind whether or not the simulation gives a good approximation. Then give them scenarios where it does approximate closely and scenarios where it doesn't as Matthew Towers described. Then they would have lots of tools... Commented Sep 27, 2021 at 17:14
• What level are the students at? 2nd year undergrads? When you say that they are "interested in applying probability", what do you mean? Do they plan on becoming statisticians? A statistician who doesn't understand the math behind expected values would not be good at his/her job. I think often there are students who aren't necessarily interested in anything except graduating while doing as little work as possible, and they don't want to solve anything if they can have a computer do it instead. However, depending on their majors, I assume that actually understanding the concepts is helpful.
– Joe
Commented Sep 28, 2021 at 1:42
• @Joe These are adults who need to use basic probability at their job: doctors, analysts, etc. Commented Sep 29, 2021 at 6:13
• This is just a nice example: jakevdp.github.io/blog/2017/12/18/simulating-chutes-and-ladders In the end this is a question about brute force vs. using more advanced tools.
– BKE
Commented Sep 29, 2021 at 20:15

Your question could apply generally to why should anyone learn the math "behind" anything, if they can easily compute the answer on a computer. I don't think they ALWAYS should. There should be a reason. For example, I had an old professor who said that when calculators first became common, some professors insisted that students should still learn to take square roots (of non square integers) by hand, in case all of the batteries died. I doubt that would be a popular opinion today.

The main reasons (not exhaustive) that I think that someone should learn the math "behind" anything is if:

1. That helps them to understand the concept better
2. That helps them build toward more advanced material that they will study later
3. They will face situations in which they cannot use a computer
4. They will face situations in which doing it without a computer is more efficient (e.g. faster)
5. That helps them apply a sanity check on the results returned by the computer

If someone is already in their chosen profession, and will not be studying anything else later, perhaps #2 doesn't apply. But, if they want to eventually model disease transmission using stochastic differential equations, then that could entail computing the expected value of quantities based on equations (not specific values of the probabilities), and hence a computer simulation probably cannot be used to determine the general solution (only solutions for particular values of the variables). That may not be the quality of analysis that could be published in medical journals.

I've had students in college algebra classes make mistakes on a calculator resulting in answers that are obviously incorrect, and when I ask, "Does that answer make sense?", they just hold up the calculator toward me. In general it can help to have some understanding of what you are doing, rather than having complete faith in the computer/calculator (and your ability to enter the correct input).

But if someone is paying to take a college course for a particular purpose, and that purpose is satisfied by learning to use simulations to compute quantities, great.

they can just run a simple simulation

Simulations outside the classroom are often anything but simple. Brute force simulations are often a lot of work to set up, take a long time to calculate, and take even more computational resources to make sufficiently accurate. It is therefore necessary to have more advance tools in one's toolbox.

Here is a very nice worked example of Chutes and Ladders.

Your question prompted me to reread a comment that John Tukey wrote in the March 1979 edition of the Journal of the American Statistical Association (pp. 121-122).

He referred to what he calls "the whole data analyst-statistician." Such a person is able to "take quite different views and adopt quite different styles as the needs change." Evidently, you want your students who are working adults to grow in these terms.

Tukey goes on to describe probability modelers as wanting to believe that their models are entirely correct, and data analysts as regarding their models as "benchmarks in a wilderness, and expecting little truth." These are two opposite ends of a spectrum. He refers to the development of "robust/resistant techniques" that will bring these two together. His "whole data analyst-statistician" is an interdisciplinary thinker who can bring these perspectives of models together. A person who can go back and forth between these two ends of the spectrum is someone who appreciates the meaning of the famous statement by George Box, who was a colleague of John Tukey: "All models are wrong, but some are useful."

Your question about bringing students to appreciate both analytic results and simulations leads us naturally to the philosophical thinkings of these two giants of statisticians (Tukey and Box). One idea for inspiring motivated undergraduates is to have them read the first seven pages of Tukey's 1962 paper "The Future of Data Analysis." I find Tukey's opening sentence to be both chilling and relevant to your predicament: For a long time I have thought I was a statistician, interested in inferences from the particular to the general. But as I have watched mathematical statistics evolve, I have had cause to wonder and to doubt.