Engaging students: Probability and odds

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place. I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course). This student submission again comes from my former student Angelica Albarracin. Her topic, from Pre-Algebra: probability and odds. How can this topic be used in your students’ future courses in mathematics or science? Probability is a topic that commonly appears in biology in the study of sexual reproduction. Both in freshman and college level biology, students are required to learn how to create and use Punnett squares. Punnett Squares are used to determine the likelihood certain alleles will appear in the offspring of 2 organisms. These alleles can do anything from determining eye color, to determining whether or not an organism will have a hereditary disease such as hemophilia. Though statistics is not a required mathematics class for high schoolers in the state of Texas, many students will end up encountering this class in high school and/or college as it pertains directly to many fields of study such as math, biology, chemistry, and physics. One of the most important concepts in statistics is the idea of statistical significance. Using the scientific method and other techniques for conducting a survey or experiment, it is easy to analyze, and record data. However, a major component of statistics is being able to interpret the implications of any given data. One of the biggest indicators that an experiment or survey that was conducted holds real implications is its statistical significance, which is essentially a measure of the probability of observing results as extreme as what was observed. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)? Speed running is a category of gaming that has become hugely popular over the years in which highly skilled and knowledgeable players compete amongst each other to complete a game as fast as possible. One of the most popular of these games in this scene is Minecraft and due to Minecraft’s popularity, speed runners of this game often come within seconds of world records, meaning every small optimization could be the difference between 1^st and 2^nd place on the leaderboards. Minecraft is a highly open and adventurous game primarily because each “world” is randomly generated, meaning that no two playthroughs are alike. This randomness not only encompasses world generation, but also factors into the availability of resources in the form of animals, enemies, and even ores used for building and crafting items. The most notorious section of the game where random generation plays a huge role in the speed run is in the collection of an essential item known as the ender pearl. In order to reach the final stage of the game, a minimum of 12 ender pearls are required, which can only be obtained from Endermen, a type of enemy in the game. Though ender pearls are considered an essential item for the completion of the game, it is theoretically possible to complete the game in its entirety without ever obtaining a single pearl.  This is due to a unique mechanic the game uses to allow the player into its final stages. Ender pearls are used in combination with a material called Blaze Powder to make a new item known as an Eye of Ender. Eyes of Ender are used to both locate a special portal to allow players into the “End” and to activate said portal. This portal (known as the End Portal) can only be activated with 12 eyes, but this is where the game’s inherent randomness plays an important factor. For each of the 12 slots in the portal dedicated to the placement of the eyes, there is a 10% chance that there will already be one inside, meaning the player would not need to provide one of their own. It is also important to note that while Eyes of Ender are used to locate this portal, it is completely possible to find this portal on your own, it is simply faster to use the Eye of Ender as a guide (and being faster is in the interest of speed runners). With this being said, the probability a player can complete the game without the usage of a single ender pearl is about 1 in 1 trillion! So, what’s the big deal? Speed runners can simply obtain the required pearls and ignore this possibility, right? Normally this would be the easy answer, but it becomes a bit more complex when we consider the nature of ender pearls. As mentioned earlier, ender pearls can only be obtained from endermen, and while their exact spawn rates are unknown, they are considered to be uncommon. In addition, each endermen has only a 50% chance of dropping an ender pearl upon defeat. If you consider this with the fact that enemies primarily spawn during the night cycle of the game, it is easy to see how obtaining these pearls can take a lot of time, something a speed runner wants to avoid at all costs. Consequently, runners are often put into a scenario in which they must balance their risk and reward. Though the probability a runner will encounter an End portal with all 12 eyes built in is near impossible, the likelihood that 2 or even 3 eyes would be there is not so low. Should a speed runner devote more time to finding ender pearls, though some of their effort maybe be for nothing, or should a runner find most of the pearls, and hope the rest are at the portal waiting? In a category of gaming where every second counts, probability can be used to figure out the most optimal answer to this question, and lead hopefuls to new world records. How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? An important concept in probability is the Law of Large Numbers which states that “the relative frequency of an outcome approaches the actual probability of that outcome, as the number of repetitions gets larger” (see link below). This law can be easily observed through repeatedly tossing a coin or rolling a die, however, as the law suggests, this must be done a large amount of times. Tossing a coin 500 times in the classroom, while helpful to demonstrate this law in action, is time consuming and tedious. As a remedy to this, Texas Instruments developed an app for TI-84 graphing calculators called Probability Simulation.

In this free app, students can choose from a variety of actions to simulate such as tossing coins, rolling dice, picking marbles, and drawing cards. In the image above, the calculator is simulating the results of rolling two die. There are many useful features and settings within this app but two of the best ones are the ability to perform an action 50 at a time (indicated by +50) and a graph to keep track of the results of all previous actions. Having the ability to perform each action quickly and in large quantity makes this a much less time consuming and material intensive activity. In addition, having a graph documenting each result from previous actions also helps tremendously in demonstrating the Law of Large Numbers as it acts as a visual aid. In the picture above, the rough formation of a bell curve can be seen after 501 rolls. References: https://education.ti.com/en/building-concepts/activities/statistics/sequence1/law-of-large-numbers https://www.minecraftseeds.co/stronghold-with-end-portal/ https://www.speedrun.com/mc/full_game#Any_Glitchless