Time is flying by as we investigate the mathematics of chance.
It’s hard to believe that Discrete Probability is half-over! Every class I’ve taught at Proof School has been fun and rewarding, but the time has never flown as quickly as it has this block; there’s so much enthusiasm for the subject in our little seminar that each two-hour meeting feels too short.
Probability was invented when gamblers consulted mathematicians on how to improve their odds. Proofniks may not be keen on gambling, but we all have our favorite games of chance, and much of our course so far has been devoted to analyzing games. We found the probability of drawing a hand with only three suits in bridge (a game some of us learned in Mr. Gregg’s Block 3 club). We worked out a baseball team’s chances of winning the World Series. We combed my very own “stats” in a trivia game to predict my chances of acing a six-question quiz, and marveled at how well the predictions matched the record. One member of the group used expected-value techniques to find the average length and preferred playing position in Umizumi, a game he learned at a math camp (and a descendant of the ancient Roman game of morra).
This was by no means the only student-created problem to date. Many serious gamers know that two 10-sided dice can substitute for a 100-sided die—just roll for the tens digit and the units digit—but can you design a spinner with the property that spinning it twice, and adding the results, simulates one spin on the spinner below?
Both this problem and its elegant solution are due to a member of the class who saw a creative way to apply the method of generating functions.
Indeed, fun problems in probability sometimes seem to condense out of thin air. At the morning community meeting two weeks ago, Dr. V led a mixer which had everyone look in a random person’s direction, peeling off the group when they made eye contact; many students were surprised at how many rounds went by with no pairs making contact, and just a couple of days later, we were able to demystify this observation in class. Another day, over my morning coffee, I checked the website FiveThirtyEight—which posts a weekly puzzle contest—and found a problem perfectly suited to the day’s lesson plan. At least one student ended up submitting his solution in the contest, along with a suggestion for an extension of the problem! And a Berkeley theater company’s unique production of Hamlet, in which seven very well-rehearsed actors draw their roles out of a skull at showtime, inspired us to figure out how many shows one would have to attend to see all seven play Hamlet. (In case you’re wondering: buy 16 tickets for a 50-50 shot at catching ’em all. By the last performance, you’ll be ready to play the role yourself.)
Students come to the Discrete Probability seminar with a range of backgrounds. Some are able to wring a little extra from the subject by using their knowledge of calculus. Some arrived with extensive skill working with infinite series; others have had to learn this skill as we go. Some have been writing proofs for years, while others are still learning the ropes. The varied knowledge of our students could have made for an atomized classroom. Instead, the opposite has happened, as Proofniks have seized the opportunity to teach each other the cool stuff they know! At the beginning of the block, given a choice between two available classrooms—the more capacious Lab or the smaller Fish Tank with its single seminar table—students voted for the more confined, intimate quarters, which set the tone for a class where conversation is the norm and everyone cares about what everyone else is doing.
In the next three weeks, we’ll model meteor showers and rains of fish, prove the Law of Large Numbers, and learn the mathematical definition of surprise. I’m excited, but I know it will be over much too fast!