probability skills

Tomorrow I start teaching my first probability class. At 8:30 am. So that’s a thing. (I’ve done 8:30 classes before, and they’re fine. In fact, I suspect upper-class students will be more amenable to them than the first-years I’ve taught in the past.)

Probability is a little different for me. It has proofs, sure—it’s a math class, after all—but it feels more like a collection of techniques than the other, more theoretical, upper-level classes I’ve taught. Those techniques are unified by a general philosophy: We can understand the long-term behavior of random processes. So I’ll be incorporating some philosophical texts along with the standard fare.

Because the theory of probability is so steeped in problem-solving, it seems like a good candidate for inquiry-based learning. I chose a classroom with a conference-room design and lots of blackboards (pictures here), so that it feels more like a collaborative environment.

Although it took me a while to realize it, since I’m not as familiar with the material, the nature of the subject also lends itself to standards-based grading. Because of previous confusion I’ve encountered when using the term “standards”, for this class I’m calling them “core skills”. A list is below. (The arrangement of topics has been heavily influenced by our textbook, A First Course in Probability, by Sheldon Ross.)

I feel like this class is a big step forward for me. I’m really going to try to let go of controlling what goes on in class through lecture, and to let (hopefully deep) exploration happen through my choices of topics and questions.

List of core skills in probability

1. Communication – Contribute to class discussion. Present neat, original written work.
2. Combinatorial analysis – Apply counting principles, in particular models involving permutations, combinations, binomial and multinomial coefficients.
3. Axioms of probability – Use definitions of sample spaces, events, and probability, together with Boolean algebra, to prove propositions. Explain meaning of the axioms of probability.
4. Conditional probability – Find conditional probability of one event given another. Explain meaning of conditional probability. Use formulas involving conditional probabilities.
5. Bayes’ formula – Use Bayes’ formula to compute conditional probabilities. Find how odds of an event change with introduction of new data. Justify potentially counterintuitive results.
6. Independence of events – Determine whether two (or more) events are independent. Explain significance of independence.
7. Random variables – Find the probability mass (or density) function and cumulative distribution function of a random variable. Explain what a random variable is and why they are important.
8. Expectation – Find and interpret the expected value of a random variable.
9. Variance – Find and interpret the variance of a random variable.
10. Discrete random variables – Create models with discrete random variables, including:
    • Binomial distribution
    • Geometric distribution
    • Poisson distribution
    • Hypergeometric distribution
11. Continuous random variables – Create models with continuous random variables, including:
    • Uniform distribution
    • Exponential distribution
    • Normal distribution
    • Gamma distribution
12. Joint distribution – Find the joint distribution function or joint probability mass (or density) function of two random variables. Find marginal distributions of the two variables.
13. Independent random variables – Determine whether two variables are independent. Explain meaning of independent random variables. Use independence to compute expectation.
14. Sums of random variables – Justify and apply linearity of expectation. Find the distribution of a sum of independent random variables by using convolution.
15. Conditional distributions – Find the conditional mass (or density) function of one random variable given another. Use conditioning to compute expectations or probabilities.
16. Covariance and correlation – Find the covariance and correlation of a pair of random variables. Explain the meaning of covariance. Compute variance of a sum of random variables.
17. Inequalities and Laws of Large Numbers – Use the Markov and Chebyshev inequalities to approximate distributions. Explain the Weak Law and Strong Law of Large Numbers.
18. Central Limit Theorem – Explain the conditions under which the Central Limit Theorem holds. Apply the Central Limit Theorem to approximate sums of random variables.