Elementary Probability

  • This lyric written, performed, and recorded in 2018 by Larry Lesser (The University of Texas at El Paso) won honorable mention in the 2019 A-mu-sing contest.  The song helps launch learning about permutations by showing how many ways n distinct objects can be ordered for the first non-trivial case (n = 3), modelling the systematic strategy of listing orderings in alphabetical order to make sure none are missed.  (Before using the song, students can be asked for their prediction – many will say 3 or 9 instead of 6.  After using the song, students can be asked to find the answer for n = 4, which is just small enough to generate by hand.)  The song also introduces vocabulary (“order”, “permuted”, “sort”) commonly used in this context.

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  • This poem (slightly revised from its publication in the January 2013 Journal of Humanistic Mathematics) was written by Larry Lesser from The University of Texas at El Paso.  It is a vehicle to discuss equiprobability bias – the misconception that all outcomes must always be equally likely.

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  • A song for teaching about the multiplication rule.  Using the popular topic among young adults of relationships, the multiplication principle is memorably illustrated by having Paul Simon's #1 hit song (which states only a half-dozen ways to leave your lover, not 50) revisited to show 50 literal paths for ending a relationship: (5 reasons for the decision) X (5 methods to relay the decision) X  (2 options for handling acquired stuff). The lyrics were written by Larry Lesser from The University of Texas at El Paso to the tune of Simon’s same-titled 1975 song.  The audio recording features vocals by Abeni Merryweather and production by Abeni Merryweather  from UTEP's commercial music program.  The song tied for second place in the 2023 A-mu-sing contest.  

    The structure of the problem in the song is similar to Exercise 3 in the progressive curriculum sequence outlined in the Spring 2024 Journal of Mathematics Education at Teachers College article “A Problem-based Curriculum to Develop the Multiplication Principle for Counting”: https://journals.library.columbia.edu/index.php/jmetc/article/view/11949/6300

     

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  • This limerick was written April 2021 by Larry Lesser of The University of Texas at El Paso to be used as a vehicle for discussing probabilities and expected values involved in playing a typical pari-mutuel lottery.  The limerick was also published in the June 2021 issue of AmStat News.

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  • This limerick was written in April 2021 by Larry Lesser of The University of Texas at El Paso to be used as a vehicle for discussing Simpson's Paradox.  The limerick was also published in the June 2021 Amstat News.

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  • A limerick to teach the inclusion-exclusion rule for finding the probability of the union of two events.  The poem was written by Marion D. Cohen from Drexel University and published in the January 2021 issue (vol 11 number 1) of the Journal of Humanistic Mathematics.

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  • A limerick to teach the addition rule for finding the probability of the union of disjoint (mutually exclusive) events.  The limerick was written by Marion D. Cohen from Drexel University and published in the  January 2021 (vol. 11, issue 1) Journal of Humanistic Mathematics.

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  • A cartoon suitable for use in teaching about risks and the problem with making post hoc comparisons. The cartoon is number 2107 (February, 2019) from the webcomic series at xkcd.com created by Randall Munroe. Free to use in the classroom and on course web sites under a Creative Commons attribution-non-commercial 2.5 license.

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  • A cartoon suitable for use in teaching about Venn Diagrams and the meaning of mutually exclusive events. The cartoon is number 2090 (December, 2018) from the webcomic series at xkcd.com created by Randall Munroe. Free to use in the classroom and on course web sites under a Creative Commons attribution-non-commercial 2.5 license.

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  • Summary: Through generating, collecting, displaying, and analyzing data, students are given the opportunity to explore a variety of descriptive statistical techniques and develop an understanding of the distinction between theoretical, subjective, and empirical (or experimental) probabilities. These concepts are developed with activities using Hershey KissesTM and may be extended to introduce the sampling distribution of a sample proportion. The activities are described in M. Richardson and S. Haller. (2002), “What is the Probability of a Kiss? (It's Not What You Think),” Journal of Statistics Education, 10(3), https://www.tandfonline.com/doi/full/10.1080/10691898.2002.11910683

    Specifics: The main activity uses Hershey’s Kisses to explore the concept of probability. Three specific sub-activities are performed such as: 

    1. Students explore the empirical probability that a plain Hershey’s Kiss will land on its flat base when spilled from a cup. 
    2. Students make predictions about the probability of an almond Hershey’s Kisses landing on its base when spilled from a cup, after having experimented with the plain Kisses.
    3. Students explore the properties of the distribution of a sample proportion to see whether the percentages of base landings have a specified distribution and whether they think that the number of Kisses tossed affects the shape or the mean and standard deviation of this distribution.

    (Resource photo illustration by Barbara Cohen, 2020; this summary compiled by Bibek Aryal)

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