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    • 2013 Publication

THE BIRTHDAY PROBLEM

Editor's Note: Sam H. wrote this for his advanced pre-calculus class.
In school, the infinite number of creative ways to understand problems is smothered in the small amount of time that we have to learn. This, I assume, has led to teaching the methods which have been proven to be most conventional and easiest to teach. While I think that teaching methods that are at a perfectly average difficulty is great, my interest in math has always led me to want to explore the other ways of understanding concepts. In exploring the most creative ways to understand problems, you expose the reality of forced perspective in mathematics. I use “forced perspective” to mean the average way that students’ brains tackle problems for the first time. By learning the same conventional methods for solving problems our whole lives, we develop a bias towards one method of attack over the infinite number of other ways. In exploring the many fascinating mathematical properties I could find, the Birthday Problem seemed to be the best showcase of forced perspective. It reveals our hidden bias towards logic and at the same time flaunts the beauty of creative mathematical proofs. The problem is best defined at this: How many random people must be in a room together before the chance of a shared birthday reaches 50%?. For the sake of this exploration, I want to give you the answer first. Mathematical proof shows that in a room with 23 random people, the chance that two people share the same birthday is around 50.7%. Though this answer seems to go against all conventional logic, the mathematical explanation is rock-solid.

A visual aid is the best method for beginning to understand this problem. Let us take a circle, representative of all possibilities of shared and nonshared birthdays. Though the comparison is not drawn to scale, it will serve it’s purpose for the time being:
Picture
By thinking about the problem like this, it gives you a hidden method of solving for P(b). If we could find the percent chance that no one shares a birthday, then we will have everything we need to solve the number of people who do share a birthday. The logic behind setting up the equation like this boils down to the simple calculation process. To find the chance that Person 1 might share a birthday with Person 2 and Person 3, it suddenly transforms into a nightmare of an equation. However, finding the other part of the equation is very simple. Imagine it like this:
Picture
Picture
Going back to our original equation of p(b)=1-p(b’), we can now solve for p(b). By taking .497 away from 1, we are left with .503, or the approximate value of p(b). This means that, off all of the possible outcomes for shared and nonshared birthdays, there is around a fifty percent chance that two people in a group of 23 will have the same birthday. This is crazy! The math seems to make sense, but it just doesn’t seem logical. I need to see more evidence, and I have designed a method to do so that does not require constant unique groups of 23 people. I am using a website that produces a given number of random calendar days within a given time period. I have generated a set of 23 random birthdays in the 2015 calendar year 10 times, and the results are reproduced below.
Picture
If we name the sets 1-10 going left to right, the following duplicates exist:
Picture
This gives us 6 birthday duplicates out of 10 possible groups, which is a 60% chance of a shared birthday. It appears as though the math was irrefutably correct. Even though there are 365 days in the calendar year, as few as 23 people can tip the scales of chance.
​

I really liked exploring this interesting and curious problem. When I was generating the birthdays, duplicates began to seem very common and it took me by surprise. I think that math like this always seems to remind me of the power of having a unique perspective, in or outside of the mathematical world.
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  • Home
  • Who We Are
  • HOW TO SUBMIT
  • Past Publications
    • 2019 Publication >
      • Scientific Research
      • Mathematical Exploration
      • Scientific Exploration
      • Computer Science
    • 2018 Publication >
      • Artistic Creations
      • Historical and Current Explanations
      • Mathematic and Scientific Exploration
      • Scientific Research
    • 2017 Publication >
      • Artistic Creations
      • Historical and Current Explanations
      • Mathematic and Scientific Exploration
      • Reactions and Responses
      • Scientific Research
    • 2016 Publication >
      • Historical and Current Explanations
      • Mathematic and Scientific Explorations
      • Scientific Research
      • Reactions and Responses
      • Artistic Creations
    • 2015 Publication >
      • Historical and Current Explanations >
        • Bell Curves
        • Birds Vs. Turbines
        • Energy in the Obama Era
        • The Future of Neuroscience
        • Gender Gap in Math
        • GMOs--Yes or No?
        • The History of Minecraft: How a Swedish Indie Game Came to Dominate the World
        • The Effect of Prozac on the Brain
        • Philae Lander's Discovery of Organic Molecules
        • Advantages and Disadvantages of Wind Turbines
        • Your Own Worst Enemy: An Overview of Lupus
        • The Methylhex Ban
        • The Effect of Lyme Disease on the Immune system
        • Infectious Mononucleosis
        • Replacing CFCs
        • The Switch
      • Mathematic and Scientific Explorations >
        • The 43rd Figure
        • The Clock
        • The Collatz Conjecture
        • Constructing a Soccer Ball
        • Determining how Ballparks Affect Batter's Ability to Create Hits
        • The Rotating Conundrum
        • Pythagorean Puzzle
        • Mathematic and Scientific Explorations
        • Kinetics Lab
        • Math in the Restaurant Business
        • Math as a Vessel for Social Change
        • Sustainability of Bottled Vs. Tap Water
        • Thoughts on the Lottery
        • Understanding Player Efficiency Rating
      • Scientific Research >
        • Communicating With Computers
        • The Mystery of Asthma
        • The Nanoscopic War Against Cancer
        • Phytochemistry
        • Solving the energy crisis with Intermediate Band Solar Cells
        • A Pain That Never Ends
        • Rapamycin Resistance
        • Ampacity of a Single Core Horizontal Cable
        • Morphological Properties of Texting Acronym Formation
        • cGAS and STING Expression
      • Reactions and Responses >
        • Can Humans Survive the Climate Crisis?
        • My Experience as a Teacher's Assistant
        • Ted Talk Responses
        • Teens For Food Justice
      • Artistic Creations >
        • Chandelier
        • Deltoidal Hexacontrahedon
        • Dodecahedron Card Trick
        • Eye of the Triangle
        • Free Radric Delantic Davis
        • The Grid
        • What Does A Randomly Composed Song Sound Like?
        • Science Wing Mural
    • 2014 Publication >
      • Cover Photo
      • Artistic Creations >
        • Art Using the Fibonacci Sequence
        • Computer Generated Architecture and Designs
        • Mathematical Landscape
        • Math Art
        • Math in Music
      • Historical and Current Explanations >
        • Algae Bio-Fuel
        • An Energy Alternative
        • Clean Energy In Transportation
        • Calorie Restriction
        • Creating Energy in the Modern World
        • Dietary Intervention Impact on Gut Microbial Gene Richness
        • Earthly Applications for NASA Technology
        • Explaining Relative Motion
        • Exploring Artificial Inteligence
        • Gamma Function
        • How Leaves Work
        • Hydrogen Fuel Cells
        • Music and Brain Development
        • Programming Calculators
        • The Science of Microsatellites
        • Sci-Fi Taser
        • Sloane's Gap
        • Sustainable Energy: Why Some Ideas Shine Brighter than Others
        • Understanding The Galvanic Cell
        • The Virus: Our Unforeseen Philosopher's Stone
        • What Are Fuel Cells and How Do They Work?
      • Mathematic and Scientific Explorations >
        • Astrocytes Expressing ALS-Linked Mutated SOD1 Release Factors Selectively Toxic to Motor Neurons
        • Big Bang
        • Dictyostelium Discoideum
        • The Future of Solar Cell Technology
        • And Many More...
      • Reactions and Responses >
        • Alternative Energy Sources, New but Unused
        • An Insight Into the Curious World of Ethnobotany
        • Challenging What We Think We Know
        • The Current State of American Education
        • Discovering New Numbers
        • Interview With an Architect
        • Life of Pi Response
        • Mathematical Art Video Commentary
        • Missing from Science Class
        • The Museum of Math
        • The Inside Scoop on a Real Mathematician
    • 2013 Publication