# Special topics for introductory probability

I am helping to design a low-level college course whose purpose is to teach critical thinking, logic, finance and probability. I have been tasked with developing the probability section. I am following chapter 4 from Bolstad's Intro to Bayesian Statistics. This goes from the definition of probability through Bayes' rule.

I was hoping to add one or two practical applications of these ideas. Types of errors is one idea. Another would be basic AI. I want things that are relevant to today yet easy enough for a student with only an algebra background to understand. Any ideas?

• "critical thinking, logic, finance and probability" Out of curiosity, may I ask for the context? I find this mixture a bit surprising. Dec 2, 2023 at 0:31
• @JochenGlueck: To me this sounds like a pretty standard "liberal arts math" survey course. Search on Pearson and you'll find a bunch of texts that hit those same beats -- e.g., ones by Bello, Smith, Angel, Miller, Blitzer, Pirnot, Lippman, etc. At my school I succeeded in having us pick the Pirnot text. Dec 2, 2023 at 4:07
• @JochenGlueck it is like Daniel said, a liberal arts survey course, checking the box for a critical thinking/logic/math course. It's meant to be a very practical introduction to quantitative reasoning in a liberal arts setting. Dec 2, 2023 at 13:00
• relevant xkcd: xkcd.com/1132 Dec 2, 2023 at 19:57
• Update: Based on @KevinPCostello's answer and my own interests and the goals of the class, I think I'm going to attempt to look at medical diagnostics, reading scientific charts/histograms and a quick swipe at a naive bayes classifier. I think this is a nice mix of topics relevant to today that will give the students an idea of how these topics are helpful in our society. Dec 3, 2023 at 0:57

A classic application of Bayes' Theorem is in medical testing, and the difference/conversion between "what is the probability I test positive, given I have the condition" vs. "what is the probability I have the condition, given I tested positive".

If you want a specific example to use, one would be the use of lateral flow (rapid) CoViD tests in the UK in 2021. The government proposed a community testing program where even asymptomatic people would be tested regularly in an attempt to limit transmission. In April 2021 (when the underlying case rate was very low) there were calls to scale back the testing because almost all of the positive test results were false positive. In January 2022 (when Omicron was in full swing), widespread testing became useful again.

Although Bayes' Theorem is never mentioned outright in the April 2021 article, the article contains enough information on underlying case rates, rates of false positives, etc. to be turned into a Bayes' Theorem example (when I taught probability this fall, I used that example as my introduction to Bayes).

• This is great! I forgot to mention that this was an example I was planning on. But I appreciate the real-life Covid data. This will definitely feature somewhere in the course! Dec 2, 2023 at 13:10
• Another medical-related one: Simpson's Paradox Dec 3, 2023 at 7:42
• I've seen many claims that people (esp. non-mathematicians) find it much easier to follow natural frequencies rather than Bayesian probability.  In particular, draw a simple 2-row, 2-column table: the rows representing e.g. ‘has the disease’ and ‘doesn't have the disease’, the columns representing ‘tested positive’ and ‘tested negative’, and the respective number of cases in each cell.  That makes it very easy and intuitive to see what's going on, and to read off probabilities etc.  So you might want to use that instead of (or as preparation for) Bayes' Theorem itself. Dec 3, 2023 at 18:59
• A variant of this that I remember from my school days: "A test for a particular banned drug is 99% sensitive and 99% specific. Assuming that 1% of the population use this banned drug, what is the probability that a person selected at random who tests positive is actually a user of the drug?" Dec 5, 2023 at 0:51
• There is a book by Gerd Gigerenzer called "Calculated Risk" that you may be interested in. Jan 20 at 1:42

One example of elementary probability is the so-called Birthday problem which asks for the probability that in a room of $$n$$ people two will share the same birthday. Sometimes formulated as a paradox that if there are 23 or more the probability exceeds one-half. Assuming your class has about that number you can try it experimentally. Whatever the result of the experiment you have a discussion topic.

• (+1) I remember our math teacher first described it and then proceeded to give us a practical demonstration. This was in the 10th grade if I recall correctly. I do not recall how it came up, it was not part of the curriculum best I remember (this was a long time ago). I clearly recall though that I was skeptical and found it non-intuitive. Our class sizes in those days were about 35 students, and sure enough, the quick empirical test revealed two people with matching birthdays in our class. Dec 5, 2023 at 0:31

You might already be aware of this one, given how famous it is, but the first thing that comes to my mind is the Monty Hall Problem. It doesn't require any fancy mathematical machinery, just a basic understanding of conditional probability, and there are a handful of factors that make it a particular fun topic to cover during class:

• It's easy to design a class activity around it -- for instance, you could put three textbooks on a desk, one of which you've inserted a sticky note behind the cover, and tell the class that if they pick the book with that piece of paper on it, then they get an extra couple percent added to their next assignment grade. (Also, you might want to tell them beforehand that once they choose the book to open, you will open a different book and ask if they want to change their decision, to make it clear that your action of opening another book will be independent of whether their initial choice is correct.) The class collectively decides on a book to open, and then you open a different book that does not contain the sticky note and ask them to collectively decide whether they want to change their decision. After giving them some time to debate, you can launch into the lesson.

• You can talk about intuition behind the result: "What if there were 100 textbooks, and I opened 98 of them that did not have the sticky note? Would you change your decision then?" Even students who don't fully grok the conditional probability aspect should be able to grasp the intuition, and the intuition might even help them develop a more tangible sense of what conditional probability is and why it's important.

• There's a lot of interesting history behind this problem that might be entertaining for the class -- apparently, even the famed Paul Erdős got it wrong and remained unconvinced of the correct choice until he saw otherwise in a computer simulation. (Thousands of other math PhDs got it wrong too and were so confident in their wrong answers that they wrote letters to a magazine that published a response advocating the correct result.)

• That's a fun example. I am familiar, but didn't think of it when I was thinking about topics. I'll definitely consider this option. Thanks! Dec 1, 2023 at 21:48
• I'd suggest that it might be more useful not to give bonus points for guessing the right book on the first try. The educational point is probably easier to get across if the students can try it over and over again, so they can try out different strategies, and maybe start to see trends in the results if you have time for enough repetitions. (Plus there's at least a 1/3 chance they just get unlucky, and it'd be kind of a bummer for them to go through the lesson knowing that they already lost their one chance at bonus points.) Dec 3, 2023 at 11:04
• I never really found point 2 that convincing, personally. For some reason. What convinced me back in the day was "Imagine you know the rules of the game, and you decide before you play whether you want to switch or not". Dec 4, 2023 at 10:19

Bertrand's Paradox is an old saw. The point is that trying to randomize an experiment is tricky since there can be different points of view.

• Fascinating! I had never heard of this before. Dec 2, 2023 at 23:34

I would look ar some of the basic six sigma literature and at doe. It is connected to all kinds of factory snd other process improvement. Very clear business connection.

I would eschew the Bayesian emphasis as that is sort of a flourish and doing for you, not optimal for a low level and business emphasis course. I have consulted in pharma manufacturing and frequentist approach was fine and still an upgrade for the middle managers doing annual product reviews versus no stats at all. And simpler for a new learner. Just picking colored balls from a bag. Along of course with basic data analysis like charting, anova, etc. This is not to say you can't cover Bayes thereom at one time and briefly. And I realize it's an interesting way to think about all probability. But more for you not for students.

And please no Monty Haul. Fun for brainy already know something types. A distraction from basic learning of neophytes.

• Thanks for the perspective. AI is my concentration, so being reminded that not everyone is interested in applications in that area is helpful. Dec 2, 2023 at 12:56
• I like the answer, but could you edit the first paragraph? It looks like some typos have snaked in, and it is hard to know which are those and which are abbreviations I am unaware of... Dec 2, 2023 at 15:26

dt688:

1. I would be very wary about being too difficult or particular, when teaching in a corporate environment. I.e. if GMers are your target audience. "What is interesting, to me" can lead to wrong choices, for the audience. I have learned this the hard way...even with beautiful, hard prepped lectures. And that I thought dumbed down enough. And I know a lot more about GM-like engineers, marketers, managers, finance types than I do about college teaching!

2. You don't need to jump off the whole Six Sigma cliff. I dislike the buzzword and the large programs. But they do have a very good approach for basic teaching to adult professionals in industry setting. Look into the "catapult training" workshops. Google it, see YT videos...but here is one link:

When I got this training (in an industrial setting), I remember how even some simple things like accuracy of measurement were revealed. Like first, they made us estimate the landing position and churn with that data. Then they allowed us to use a carbon paper that marked the location and so got more accurate data. This is something that clearly makes sense in the manufacturing environment. But is not drummed in, in the average academic stats course, with lots of equations and the like. Remember Pearson (or Dewey or whoever) who said to his stats grad students, "Never forget that your municipal data came from the night watchmen."

3. Of course, you are a quality engineer, and thus know the drill.

Tommi:

DOE, Six Sigma and ANOVA should be capitalized. And "snd" is "and".

I would suggest geometric probability and applications in stereometry!

• This is outside of my area of expertise. It looks like it might be beyond the scope of this course as well. Thanks though! Dec 3, 2023 at 0:54