Q&A: Activity for a Probability Class

Cross-posted from here.

Question

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. Any ideas?

Answer

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.)