Math Digests November 2022

Everything you need to know about the math of Powerball

Big Think, November 7 2022

This month, one lucky person won a record-setting \$2.04 billion in the Powerball lottery, beating unthinkable odds of 292 million to 1. Surely that \$2 billion winner feels their \$2 lottery ticket was a good idea. But would a mathematician have advised they buy the ticket, not knowing they’d win big? There’s more to the math of Powerball than just the odds of matching five white balls numbered 1 through 69 and one red ball from 1 to 26. “That probability doesn’t factor in the cost of a ticket versus how much you can expect to actually take home: the mathematical definition of ‘expectation value’,” writes Ethan Siegel, in an article for Big Think. In this article, Siegel breaks down the math behind the question of when exactly playing Powerball is “worth it” mathematically. 

Classroom Activities: probability, expected value

  • (Mid Level) Based on this article, if a lottery ticket costs \$5 and you have a greater than 50% chance of winning \$9, is playing worth it? 
    • What if you have a greater than 1% chance of winning \$500?
  • (High Level) Suppose the lottery drawing consists of two white balls numbered from 1-50, and one red ball numbered 1-20.
    • What is the new probability of matching all three balls for the jackpot?
    • What is the new probability of matching one white ball and the red ball?
    • How big must the jackpot prize be for the overall expected value to surpass the \$2 ticket cost?

—Max Levy

What Students Are Saying About the Value of Math

New York Times, November 10, 2022

What’s the point of learning math? Your teachers, parents, and peers may have instilled in you the value of learning math—to adjust baking recipes, to design or create art, to solve problems in science and engineering. But do you feel that your higher-level math education in algebra, geometry, statistics, or calculus is worth the effort? A recent article for the New York Times shares how students around the country responded to this question. Some answers are positive: “Math is timing, it’s logic, it’s precision, it’s structure, and it’s the way most of the physical world works.” Others are more negative: “Math could shape the world if it were taught differently.” Reading the spectrum of responses can help us find ways to appreciate math education, as well as ideas for how to improve it.

Classroom Activities:  algebra, geometry, calculus, education, math anxiety

  • (All levels) Discuss the writing prompts from this survey in groups of three or four. Based on your discussion, write a paragraph individually answering the question: Is it important to learn math in school? Why or why not?
  • (All levels) Several students responded to the prompt with negative responses about math. One wrote, “I think math is a waste of my time because I don’t think I will ever get it.” Respond to the following prompts:
    • Can you remember a time that you struggled with a subject or activity you enjoy? What helped you overcome that obstacle?
    • What would you change about how math is taught at your school?
    • Math is essential for understanding concepts in science and engineering. Is it useful for people who don’t want to work in these fields? Why or why not?

—Max Levy

Strategic voting is possible but risky on a ranked choice ballot, mathematicians say

Alaska Beacon, November 5, 2022

After the death of their Representative in the US House, Alaska held a special election in August. The results were surprisingly interesting. Mathematicians Adam Graham-Squire and David McCune showed that if certain voters had switched from a losing candidate (Sarah Palin) to the winner (Mary Peltola), that influx of votes would have made Peltola lose, as James Brooks writes for Alaska Beacon. In other words, more votes need not be better. This counterintuitive outcome, called the monotonicity paradox, was possible because the election used ranked-choice voting. In ranked-choice voting, voters rank candidates in order of preference. The candidate with the fewest first-place votes is eliminated, their votes are reassigned to those voters’ second-favorite choices, and the process repeats until someone has a majority of votes. If some of Palin’s votes were transferred to Peltola, then Palin would have been eliminated in the first round. Peltola would have then lost in a head-to-head comparison with the third candidate, Nick Begich III.

Classroom Activities: voting, elections

  • (All levels) Suppose that Candidate A, Candidate B, and Candidate C are running in an election, and that 40 voters prefer A, then B, then C; 35 voters prefer C, then B, then A; and 25 voters prefer B, then C, then A.
    • Who would win if the election were awarded to the candidate with the most first-place votes?
    • Who would win in the ranked-choice voting system described above?  
    • If Candidate C dropped out of the race, who would win in a race between just A and B? Who would win in a race between just A and C, or a race between just B and C? (A candidate that would win in a direct comparison with any other candidate is called a Condorcet winner. In the Alaska election, Begich was a Condorcet winner but lost the election.)
    • For more information about different voting systems and how they can lead to different outcomes, watch this video from PBS Infinite Series.
  • (All levels) The United States Electoral College is another example of a nonstandard voting system. Read the article The Electoral College, According to a Math Teacher by Ben Orlin. Try exercises 1-10 under the heading “Electors Per Capita: What Does It Tell Us?”
    • Pair up with another student and compare your answers, especially for problems 5, 8, 9, and 10.
    • Now, discuss the article as a class. Share your thoughts on the following questions:
      • What did you learn about the Electoral College that you didn’t know before?
      • Did the article and exercises change your opinion on the Electoral College? If so, how?
      • Is the Electoral College fair? Why or why not?

—Tamar Lichter Blanks

Meet Ada Lovelace, The First Computer Programmer

Discover Magazine, November 3, 2022.

Ada Lovelace was a nineteenth-century mathematician known especially for her collaboration with Charles Babbage on early prototypes of computers. Some of her most groundbreaking work appeared in a paper she wrote about a machine Babbage had conceived of, called the Analytical Engine. Babbage envisioned that people would give the Analytical Engine instructions about formulas and values to compute, and it would print out the requested information. In her paper, Lovelace laid out a procedure for calculating Bernoulli numbers on the Analytical Engine. But some have held onto suspicions that Lovelace’s work was truly original. For Discover Magazine, Emilie le Beau Lucchesi details Ada Lovelace’s accomplishments, as well as the sexist barriers that sprang up both in life and after her death.

Classroom Activities: programming, recursive sequences

  • The Bernoulli numbers form a recursive sequence, meaning they form an infinite list, and each number in the list depends on the previous items.
    • (Mid level) Try Khan Academy’s lessons on recursive sequences and do the practice problems.
    • (Mid level) Think of a recursive formula of your own, and calculate the first 5 terms in your sequence.
    • (High level) Read this Project Lovelace page describing Ada Lovelace’s Bernoulli number algorithm. After learning what Bernoulli numbers are and how to calculate them, calculate the first 5 Bernoulli numbers.
  • (High level, programming knowledge required) Write a computer program that, when given a number n, outputs the nth element of the recursive sequence you came up with in the last exercise.

—Leila Sloman

Fireflies Sync Up Their Dazzling Light Shows With Mathematical Precision, Scientists Find

Popular Mechanics, November 4, 2022.

A large group of fireflies flashing in the night can be delightful to watch, especially if you’re not used to seeing them. But sometimes something even cooler happens: The entire group coordinates their flashing. This isn’t common, writes Tim Newcomb for Popular Mechanics, but it was mysterious enough for mathematicians to explore further. The flashing fireflies form a dynamical system, a system that evolves over time, and their coordination is a dynamical phenomenon called “synchronization”. The researchers (Madeline McCrea, Bard Ermentrout, and Jonathan Rubin) simulated flashing fireflies in a mathematical model, which they refined until they observed the synchronization they were looking for.

Classroom Activities: dynamics, differential equations

  • (All levels) At the start of this talk, mathematician Steven Strogatz gives several real-world examples of synchronization similar to what the fireflies exhibit. Watch starting at time 1:47 until minute 14:00 to see him describe all of the examples.
  • (High level, Calculus) When modeling dynamical systems, mathematicians often use differential equations. These are equations that involve a function f and its derivatives. Typically, a “solution” to a differential equation means a function f(x) that satisfies the equation. For instance, f(x) = 2x satisfies the differential equation f’(x) = 2. In the following exercises, show that f(x) solves the given differential equation.
    • f(x) = e^x; \; f'(x) = f(x)
    • f(x) = e^{x^2}; \; f'(x) = 2xf(x)
    • f(x) = x^3; \; f'(x)^3 = 27f(x)^2
    • f(x) = \sin(x); \; f''(x) = -f(x)
    • (Integral Calculus) Find a solution f(x) to f'(x) = f(x)^2. (Hint: Use u-substitution.)
  • For more on differential equations, and some harder examples, check out Paul’s Online Notes.

—Leila Sloman

Some more of this month’s headlines