# Teaching undergraduate probability

When teaching undergraduate probability, I post additional material for students to read. Many students and some of our faculty find it interesting. Below are the topics with the references that I post. The textbook used in class is “A First Course in Probability” by S. Ross, 8th edition. If you want to suggest any topics to add to the list, feel free to email me.

**Sudoku**:

“The science Behind sudoku”, J.-P. Delahaye, *Scientific American*, (2006), June.

This is a fun Scientific American article about Sudoku and underlying questions in combinatorics. A number of most basic combinatorial problems related to Sudoku remain unsolved!

**Venn diagrams**:

“The search for simple symmetric Venn diagrams”, F. Ruskey, C. D. Savage, and S. Wagon, *Notices of the AMS*, (2006), Volume 53, Number 11, pp. 1304-1311.

This is an article on Venn diagrams from Notices of the AMS (American Mathematical Society). A basic question is how to draw Venn diagrams and how to think about them for the intersections of more than three sets. At least look at some fascinating pictures!

**Tossing a coin**:

“The probability of heads”, J. B. Keller, *The American Mathematical Monthly*, (1986), Volume 93, Number 3, pp. 191-197.

No other probability model is as simple as that for tossing a coin. But “Why is the outcome of a coin toss considered to be random, even though it is uniquely determined by the laws of physics and the initial conditions?” The posted article looks into this and related questions by analyzing the motion of a tossed coin. Assuming that the motion is determined by an upward velocity u and an angular velocity w of the coin, a starting problem is to determine what values of these velocities result in the coin landing heads or tails. Fig. 2 plots the regions for these values which, interestingly, have an alternating pattern.

**More on birthday problem**:

“The birthday problem”, M. C. Borja and J. Haigh, *Significance*, (2007), September, pp. 124-127.

This is an article with slightly more information on the well-known birthday problem discussed. It discusses what happens when the birthdays are not distributed uniformly across the year, and also mentions a number of other interesting generalizations of the problem.

**More on the Monty Hall problem**:

“The car and the goats”, L. Gillman, *The American Mathematical Monthly*, (1992), Volume 99, number 1, pp. 3-7.

In the so-called unconditional version of the Monty Hall problem, the probability of winning is 2/3 if you switch. The posted article discusses, among other things, the conditional version of the problem. In this version, the probability could be a number from 1/2 to 1. You can also see Wikipedia for more information on the problem.

**(Conditional) Probabilities in the O.J. Simpson trial**:

“When batterer turns murderer”, I. J. Good, *Nature*, (1995), June, Volume 375, p. 541.

“When batterer becomes murderer”, I. J. Good, *Nature*, (1996), June, Volume 381, p. 481.

Here is a nice article on conditional probabilities from a NYT blog. It talks, in particular, about conditional probabilities in the O. J. Simpson trial. I post here the two referenced correspondences related to the case which appeared in Nature.

**The case of Sally Clark**:

“Multiple sudden infant deaths – coincidence or beyond coincidence?”, R. Hill, *Paediatric and Perinatal Epidemiology*, (2004), Volume 18, pp. 320-326.

“Reflections on the cot death cases”, R. Hill, *Significance*,(2005), March, pp. 13-15.

This concerns a famous case in the UK where a mother (Sally Clark) was convicted of murdering her two newborns. The two deaths were sudden and unexplained. The trial is also famous for misuse of independence and conditional probabilities (the so-called Prosecutor’s Fallacy). Here is one easy-to-read article about the case. The article was written before Clark was released after the second appeal. The term “cot death” used in the UK is the same as SIDS, Sudden Infant Death Syndrome, used in the US. The other two posted articles also summarize and discuss the case.

**Hot hand, gambler’s fallacy and other misperceptions of randomness**:

“Probabilities of Clumps in a Binary Sequence (and How to Evaluate Them Without Knowing a Lot)”, D. M. Bloom, *Mathematics Magazine*, (1996), Volume 69, Number 5, pp. 366-372.

“The cold facts about “hot hand” in basketball”, A. Tversky and T. Gilovich, *Chance*, (1989), Volume 2, Number 1, pp. 16-21.

“The perception of randomness”, M. Bar-Hillel and W. A. Wagenaar, *Advances in Applied Mathematics*, (1991), Volume 12, Number 4, pp. 428-454.

It is known that people often misperceive randomness and probabilities. Remember the birthday problem? Another famous example is that of runs or streaks in independent trials, which are much more common than many people believe. For example, the probability of a run of three or more heads in 10 independent tosses of a fair coin is about 0.5. In one of the posted articles, it is shown how probabilities of such runs can be computed.

From a different angle, this also means that real-life phenomena exhibiting runs may actually be consistent with the hypothesis of independent trial. For example, in another posted article, it is argued that there is no such thing as a “hot hand” in basketball. (There is definitely no such thing in gambling, though many people still believe it).

Finally, the third posted article provides a review on (mis)perceptions of randomness. It may be a bit more advanced.

**Monopoly and Markov chains**:

“How fair Is monopoly?”, I. Stewart, *Scientific American*, (1996), April.

“Monopoly revisited”, I. Stewart, *Scientific American*, (1996), October.

Quoting one of the articles: “Everyone has played Monopoly. But few, I’d imagine, have ever thought about the math involved. In fact, the probability of winning at Monopoly can be described by interesting constructions known as Markov chains.” Markov chains are discussed in Section 9.2 of the textbook. Perhaps you even had them in another course. To understand the posted articles, you will likely have to read first about Markov chains.

**Card counting**:

Do you know what card counting is? Do you know that this has little to do with memorizing cards? Do you know who invented it? Do you know that the same person is considered one of the first quants (of Wall Street)?

There is a fascinating interview with the inventor of card counting (Ed Thorp) and Scott Patterson, the author of the book “The Quants”, here. You can read more about card counting on Wikipedia or just by googling it. Perhaps as a possible reference for your future, there is even a serious probability book on the subject of gambling “The Doctrine of Chances: Probabilistic Aspects of Gambling” by Stewart N. Ethier, Springer, 2010.

**Card shuffling**:

“How many times should you shuffle a deck of cards?”, B. Mann, Ubpublished.

There is an interesting application of probabilities to card shuffling. The basic question is how many times a deck of cards needs to be shuffled (the number is 7). The story made it to the popular culture, for example, here. A fairly accessible (but not easy) article on the subject is posted, with references to the original work.

**Coincidences and probabilities**:

“Methods for studying coincidences”, P. Diaconis and F. Mosteller, *Journal of the American Statistical Association*, (1989), Volume 84, Number 408, pp. 853-861.

Quoting the posted article:

“Coincidences abound in everyday life. They delight, confound, and amaze us. They are disturbing and annoying. Coincidences can point to new discoveries. They can alter the course of our lives; where we work and at what, whom we live with, and other basic features of daily existence often seem to rest on coincidence.”

How does one think about coincidences through a probabilistic lens?

Here is an easy to read article on this. A more technical article relate to the subject is posted.

**On the foundations of Probability Theory**:

“What is a random sequence?”, S. B. Volchan, *The American Mathematical Monthly*, (2001), Volume 109, pp. 46-63.

Posted is an article on some alternative approaches to Probability Theory. It involves a number of more complex ideas (e.g. computing), and may be too advanced for undergraduates. At least, one should be able to make it through the introduction.

**Power-law distributions**:

“Cascading failures: extreme properties of large blackouts in the electric grid”, P. D. H. Hines, B. O’Hara, E. Cotilla-Sanchez, and C. M. Danforth, 2011.

This is an article on power-law distributions. It appeared as part of the Mathematics Awareness Month, April 2011. That year the awareness month was on “Unraveling Complex Systems”.

**Optimal stopping**:

Here is a nicely written article on the so-called optimal stopping. The marriage problem discussed in the article appears as Example 5k in the textbook, p. 344.