By Dr. Lawrence Lesser
Since April is Mathematics Awareness Month and this year’s theme is “Mathematics, Magic, Mystery,” let’s take a look at how statistics is magical, too!
Persi Diaconis went from professional magician to professor at Stanford University, rigorously solving questions such as how many riffle shuffles put a deck of playing cards in an order that can be viewed as random (the answer: 7) and co-authoring the book Magical Mathematics: The Mathematical Ideas that Animate Great Magic Tricks.
Michael Posner of Villanova University and Boston University’s Mark Glickman each have performed statistics-related magic tricks at the United States Conference on Teaching Statistics (a video of Glickman’s performance will be posted soon in the Fun resources collection of the Consortium for the Advancement of Undergraduate Statistics Education website).
Beyond mere attention-getters, magic tricks can be an innovative, visual and memorable vehicle to explore concepts in a probability and statistics class. Glickman and I published in 2009 a comprehensive overview that Model Assisted Statistics and Applications kindly made freely available.
In our paper, the tricks-statistics content connections include basic probability and combinatorics, probability and sampling distributions, hypothesis testing as well as more advanced topics such as Markov chains and Bayes’ Theorem. Some tricks require special props or practice; some don’t.
In addition to connecting with specific content, we find magic an effective way to engage students’ critical thinking about assumptions (e.g., was that selection random?)—a key statistical habit of mind. Let me illustrate with a trick I do when teaching the introductory statistics course. (In class, I say “demonstration” rather than “trick”, so I do not lead students to expect something out of the ordinary.)
After asking if anyone in the class has ESP (no one raises their hand), I announce we will test ourselves by making predictions about the outcome of each of the following four situations:
- a card from a 52-card deck
- a roll of a six-sided die
- a spin of a spinner with five equal-sized, differently-colored regions
- a flip of a coin
Each semester I do this, I am the only one in the room who is successful with all four predictions (thanks to the trick described in our paper) and we validate the class’ collective gasp at my feat by applying the recently learned probability rule known as the multiplication rule for independent events. By making the assumption of independence, the class verifies that getting all four predictions correct indeed has an impressively small probability: 1/52×1/6×1/5×1/2 = 1/3120.
This exercise motivates discussion about key concepts of hypothesis testing, such as what is the threshold for how unlikely an observed event would need to be to reject the hypothesis that it was simply chance. After all, if I had instead only flipped a coin and correctly predicted the outcome, everyone rightfully would have been totally unimpressed (since such a feat would happen just by chance about half the time).
And so, statistics is magical—not as in some elusive black-box fashion—but as a powerful tool to illuminate the unknown and captivate our students.
Lesser is a professor in the Mathematical Sciences Department and interim director of the Center for Excellence in Teaching and Learning (CETaL) at The University of Texas at El Paso.