Matrix Algebra Plinko Probability Problem

Towards the beginning of Linear Algebra, we came across this problem:

Seeing those decimals, I realized (or remembered) matrices could be used to model probabilities, fractions, etc. I’d already learned on my own about Markov Chains and the like, so I pretty much knew how they worked. I began working on a problem on my own, one which modeled a Plinko board.

A diagram of the modeled Plinko board.

I finished a rough version of the matrices themselves that very period. I showed it to my teacher towards the end, and he said I was on the right track. We spoke for a bit about Markov Chains, and then the period was over.

I obsessed over the problem for the rest of the day. During lunch, I polished the probability matrices, making sure each entry made sense. With some down time in Economics, I solved the problem I’d created, and played around with it to my satisfaction. With nothing else to do, I let it lie. It almost ended there.

Then, just two weeks ago, we learned about probability matrices in class. I remembered the Plinko problem I’d devised all that time ago. I dug it out of my bag, looked at the work, and realized I could have a viable practice problem. It was the class before spring break, and my teacher said we’d be doing review the class after. I asked whether, if I typed up the problem in LaTeX (which I did and still do have difficulty with, thus allowing for a learning experience), he or I could use it as a class example during the review. He said yes, and I looked forward to doing it over the break.

I’d wrestled with LaTeX before, and had even attempted to type up this exact problem on a prior occasion. Usually my attempts ended in failure, for a variety of reasons. LaTeX is almost entirely unlike everything else I’ve worked with, and partakes in next to zero programming (so to speak) conventions. Everything needs to be looked up – you can’t get anywhere using common sense and intuition. And that leads into the main problem – documentation is incredibly hard to find. Google searches sometimes yield results, but are oddly sporadic in what they do and don’t cover. Books are practically nonexistent in bookstores both specialized and general, and digitally available titles range from eBook only textbooks with $500 licenses to presumably illicit PDF files consisting of only the first two chapters of an in-depth book to Amazon results usually either unsatisfactory or… impure. I’ve not had much luck.

To make sure this went differently, I asked my teacher for some resources on LaTeX, including the files for our class notes, which would prove to be invaluable for providing examples. He also provided a recommendation for a TeX front-end program, TeXworks, and an 880 page digital manual for a graphics package, TikZ. To be honest, I found simpler alternatives to both that worked just fine, but I was still impressed with the complexity and power of each.

Typing up the problem was simpler than I expected it to be. Ironically, with the lessons I’d learned in previous attempts, and the resources on my side, I encountered more trouble making satisfactory examples than technical trouble. The biggest problem I encountered (text turning italic after a subscript due to math mode interpretation) was both avoidable and well documented. But I wanted to make good, semi-challenging examples that’d teach the class something new, and while I feel like I succeeded at teaching something new, I never found a relevant operation more challenging than simple matrix multiplication. I may try to improve it in the future.

While we’re on the topic of potential improvements (and difficulties in designing the worksheet), I had trouble deciding whether to add a second part my teacher suggested which I really liked – a mathematical model for a hypothetical, infinitely tall Plinko board. If you had the probabilities between each subsequent row all the way down, you could determine the probabilities of which slot the ball/puck would fall into at the hypothetical bottom of the board.

Anyway, I finished the worksheet, sent it to him, and let it lie until the end of break, which happened to be today, as I write this. I got in this morning, he and I talked about it for a while, and then he said I could do the worksheet for the class right after the review. That’s the plan we went with.

Teaching the examples themselves went all right. I feel like it left something to be desired, but that’s probably me generalizing trouble with the SmartBoard to my performance in the whole thing. However, the experience did show me that while I’m very good at one-on-one instruction, I still need to build matching skills in teaching a whole classroom at once. Not sure how to achieve this, but I’ll try to figure something out. In terms of enjoyment, I probably would’ve been more satisfied helping out my teacher with working through the problem at the front, and providing commentary on points of interest/confusion/contention.

I think that’s everything. Thank you for reading, and the worksheet is below. The (admittedly rough) LaTeX is available on request!

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