Pure Mathematics

Wikipedia defines mathematics as "the study of quantity, structure, space and change." With a definition like that, it's easy to see why mathematics is often called "the language of science." Mathematics is essential for analyzing and communicating scientific results, and for stating scientific theories in a way that is clear, succinct, and testable.

If you don't find something that grabs your interest, keep checking back, because we'll be adding new projects in other areas of math!

Additional Project Ideas

  • Money Problems
    Math can make you money! If you understand some basic math, you can make good decisions about how to keep, spend, and use your hard earned dollars. Try an experiment comparing the same balance in different types of bank accounts. How much better is a savings account than a checking account? What difference does the interest rate make? Which is better, an account that earns compound or simple interest? Can you compare the short and long term costs of borrowing money compared to saving the cash for a purchase? (ING, 2006) Use mathematical arguments to answer classic questions like, "Which is more: one million dollars, or one penny the first day, double that penny the next day, then double the previous day's pennies and so on for a month?" (Dr. Math, 2006)

  • Scale Models
    Many industries rely on scale models to develop new products and designs. Architects, industrial designers, artists, clothing designers, and car manufacturers all use scale models. Each model is built to a scale that relates the actual object to the model through a ratio. Can you determine a formula for constructing a scale model? You can use your formula to make a model of your house, school, neighborhood, or town (CUBE, 2002). You can make scale models of the Wright Brothers aircraft designs from 1900-1903. (Storm and Benson, 2003) You can also do the opposite, blow a model up into a life size object. Can you determine a formula to convert a model into life size? You can use your formula to evaluate if a model is realistic. Try comparing toy cars, dolls, dinosaurs, or action figures to life size objects. Are the models realistic? (McCoy, 2004; Fassett and Millbyer, 2006)

  • Data Models
    Math is used by many different types of scientists to model phenomenon and evaluate data from an experiment. By building mathematical models scientists can understand how different physical, chemical, and biological processes are affected by different variables. The most important tools are: making a graph to give a visual representation of the relationships between your variables and making an equation to give a way of computing the relationships between your variables. Find a source of data, either from an online database or from your own experiment, and make a mathematical model. Try to make a graph and an equation for your data. How good is it at predicting additional results? Is the relationship linear, dynamic, or exponential? Is your data positively or negatively correlated? Are there any outliers and how do you decide if you should get rid of them? Can you show how to use a graph to evaluate sources of error in an experiment? Can you use your model to propose a hypothesis or theory for the relationship? (Wattenburg, 1998)

    Wattenburg, F., 1998. "Mathematical Modeling in a Real and Complex World," Mantana State University, Department of Mathematics. [accessed: 5/1/06] http://www.math.montana.edu/frankw/ccp/modeling/topic.htm

  • Topologies
    What do knots, maps, mazes, driving directions, and doughnuts have in common? The answer is topology, a branch of mathematics that studies the spatial properties and connections of an object. Topology has sometimes been called rubber-sheet geometry because it does not distinguish between a circle and a square (a circle made out of a rubber band can be stretched into a square) but does distinguish between a circle and a figure eight (you cannot stretch a figure eight into a circle without tearing) (Wikipedia contributors, 2006). A common joke is that topologists are people who don't know the difference between a coffee cup and a doughnut. A project in topology can have many forms. Can "Euler's Solution" help you efficiently run your errands? Can you figure out the number of possible routes to get to school? What is the minimum number of colors needed to color in a U.S. map so that no two states that are touching have the same color? Use topology to untangle knots or to discover knots that cannot be untangled. Use topology to solve mazes, draw circuit diagrams, make phylogenetic trees, or fold origami! The possibilities are endless... (Britton, 2006)

  • Fractals
    A fractal is, "a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced/size copy of the whole" (Mandelbrot, 1982). There are many different fractal patterns, each with unique properties and typically named after the mathematician who discovered it. A fractal increases in complexity as it is generated through repeated sets of numbers called iterations. There are many interesting projects exploring fractal geometry that go beyond the Exploring Fractals Science Buddies project listed above. For example, can you show that the perimeter of the Koch Fractal increases infinitely even though the area is finite? (Lanius, 2004) Can you use fractal geometry to investigate and model objects from nature? Try matching up a fractal pattern with the natural geometry of clouds, ferns, flowers, shells, or trees (Frame and Mandelbrot, date unknown). Can you program a java applet to draw different fractals? (Devaney, 1999)

  • Origami
    How do you turn a 2-dimentional piece of paper into a 3-dimentional work of art? Origami, the classical art of Japanese paper folding, is loaded with mathematical themes and concepts. What are the common folds in origami, and how do they combine to create 3-dimentional structure? Can you classify different types of origami into classes based upon the types of folds they use? Can you show Kawasaki's Theorem, that if you add up the angle measurements of every other angle around a point, the sum will be 180? (Andersen, 2004) Can you show that Huzita's Axioms, a set of seven rules of paper folding, are true? (Wikipedia contributors, 2006)

  • Playing Games
    Almost all of the games we play are based on math in some way or another. Card games, board games, and computer games are designed using statistics, probabilities, and algorithms. Begin by reading about games and game theory. Then you can choose your favorite game and investigate the mathematical principles behind how it works. Can combinatorial game theory help you to win two-player games of perfect knowledge such as go, chess, or checkers? (Weisstein, 2006; Watkins, 2004) In a multi-player game like Dots and Boxes, does the first player have an advantage? Can you formulate a winning strategy against an opponent? In single-player games can you beat the game with an algorithm? Can you invent your own game using mathematical concepts? Can you describe the math behind traditional or cultural games? (McCoy, 2004) Artificial Intelligence: Teaching the Computer to Play Tic-Tac-Toe.

  • Playing Music
    Music has many mathematical elements in it: rhythm, pitch, scale, frequency, interval, and ratio. There are many ways to turn these elements into a science fair project. You can investigate how the scale is based upon a special type of number sequence called a Harmonic Series. Another scale used by Bach, called the "Well-Tempered-Scale" or the "Equal-Tempered-Scale", is based upon a series. How are these mathematical series and ratios related to notes, chords, intervals, and octaves? You can show how these concepts are applied to generating discreet notes and pitches with different instruments. You can also investigate rhythm and time in music: can you find a way to represent your favorite rhythms numerically? (Rusin, 2004)

    Rusin, D., 2004. "Mathematics and Music," Northern Illinois University, Department of Mathematical Sciences. [accessed: 5/1/06] http://www.math.niu.edu/~rusin/uses-math/music/index.html

  • Solving Logic Problems
    You may know Lewis Carroll as the author of Alice in Wonderland, but did you know that in real life he was a mathematician who studied symbolic logic and logical reasoning? How can math help you solve Lewis Carroll's Logic Game? (Bogomolny, 2006) How are algorithms for solving the game Sudoku similar to solving a logic problem? (Hayes, 2006) For the super-advanced mathematical genius, try to evaluate currently available, logic-based computational tools, or design a better one! (SharpToolBox.com, 2006)

  • Magic Squares
    A magic square is an arrangement of numbers from 1 to n2 in an n x n matrix. In a magic square each number occurs exactly once such that the sum of the entries of any row, column, or main diagonal is the same. You can make several magic squares and investigate the different properties of the square. Can you make an algorithm for constructing a Magic Square? Can you show that the sum of the entries of any row, column, or main diagonal must be n(n2+1)/2? Are there any other hidden properties of a Magic Square? Show the differences between special instances of the Magic Square, like the Lo Shu, Durer, Ben Franklin, or Sator Magic Squares. Can magic squares be constructed in 3 dimensions? You can also investigate other shapes, like magic circles and stars (Alejandre, 2006; Pickover, 2002). Or test the question, "Is there really no math in Sudoku?" (Hayes, 2006)

  • Around the World: The Geometry of Shooting Baskets
    Take shots at a set distance from the basket, but systematically vary the angle to the backboard. For a basic project: How do you think your success rate will vary with angle? Draw a conclusion from your experimental results. A bar graph showing success rate at different angles can help to illustrate your conclusion. For a more advanced project: Use your knowledge of geometry and basketball to come up with a mathematical expression to predict your success rate as a function of angle (measured as a percentage of your success rate with straight-on shots)? How well does your prediction agree with your actual results? (idea from Goodstein, 1999, pp. 103–105.)

    Goodstein, Madeline. Sports Science Projects: The Physics of Balls in Motion. Berkeley Heights, NJ: Enslow Publishers, 1999.

  • Geometry of Goal-Scoring
    Block off one-third of a soccer net with a cone, 5-gallon bucket or some other suitable object. Shoot into the smaller side from a set distance, but systematically varying the angle to the goal line. Take enough shots at each angle to get a reliable sample. How does success vary with angle? For a basic project: How do you think your success rate will vary with angle? Draw a conclusion from your experimental results. A bar graph showing success rate at different angles can help to illustrate your conclusion. For a more advanced project: Use your knowledge of geometry and soccer to come up with a mathematical expression to predict your success rate as a function of angle (measured as a percentage of your success rate with straight-on shots)? How well does your prediction agree with your actual results? (idea from Gardner, 2000, pp. 108–110.)

    Gardner, Robert. Science Projects About the Physics of Sports. Berkeley Heights, NJ: Enslow Publishers, 2000.

  • Statistical Significance: Using a t-Test
    Sunspot activity has been monitored continuously since about 1700. The historical data shows that sunspot activity rises and falls in a roughly 11-year cycle. This project shows you how you can use a spreadsheet program to perform both graphical and statistical analysis to look for patterns in cyclical data. You'll learn how to use a t-test, which is a measure of statistical signficance. Sunspot Cycles.

  • Frequency Histograms
    Can you remember what the weather was like last week? Last year? Here's a project that looks at what the weather was like for over a hundred years. You'll use historical climate data to look at moisture conditions in regions across the contintental U.S. You'll use a spreadsheet program to calculate the frequency of different moisture conditions for each region and make graphs for comparison. Which part of the country has the most frequent droughts? The most frequent periods of prolonged rain? The most consistent precipitation? Here's one way to find out. Dry Spells, Wet Spells: How Common Are They?

  • Relationships Between Variables: Using Correlation and Linear Regression
    Here's a project that combines sports and math. You'll learn how to use correlation analysis to choose the best team batting statistic for predicting run-scoring ability (Albert, 2003). You'll also learn how to use a spreadsheet to measure correlations between two variables. Which Team Batting Statistic Predicts Run Production Best?

Resources

Sources for Additional Project Ideas

Math Links

Exploring Fractals

Objective

This project has three objectives:

  1. Explore. You should explore the complex patterns generated at the edges of the Mandelbrot set, and the relationship between a point in the Mandelbrot set and the corresponding Julia set.
  2. Understand. You will need to understand how an image of the Mandelbrot set is generated, and how the image of a Julia set corresponding to a point in the Mandelbrot set is generated. You should also understand the relationship between different points in the Mandelbrot set image and the characteristics of the Julia sets associated with each type of point.
  3. Explain. You should be able to explain in your own words what you learned in numbers 1 and 2.

Introduction

What are fractals? Here's an often-used example. Imagine that you want to measure the length of a coastline. Pick any coastline you want, but for this example we'll use the coast of California. There are different ways you could go about making the measurement. For instance, you could take measurements from a map, or you could set off on a family vacation to drive along the whole coast and measure it that way. The first idea is certainly easier, so let's consider that one first.

You'll have to decide on which map to use, and also how to measure the squiggly coastline on that map. Let's say you decide to use a map of the continental United States, and for following the squiggly coastline you're going to use a pair of dividers (like a compass for drawing circles, but with two points instead of a point and a pencil). You use the map scale to adjust the points of your dividers to span 100 km, then you "walk" the dividers along the coastline on the map—from just north of Tijuana, Mexico, to just south of Brookings, Oregon—counting the steps your dividers make. You multiply the number of steps by 100 km, and you've got your answer, right? But wait, along the way you noticed that, with a spacing of 100 km, your dividers didn't quite follow all of the promontories and bays accurately. So you reset the dividers to 50 km, and measure again. This time, you get a bigger number. Which one is right? And then you start thinking, "Hmm, maybe to be really accurate, I need a more detailed map." So you get a map of just California, and you measure yet again, and get an even larger number. You begin to realize that the amount of coastline you measure depends on the size of your ruler.

Fractals are mathematical functions that behave in a similar way. As you "zoom in" on a fractal boundary, you find that each new level of magnification continues to show complex structure. No matter how much you magnify, you never reach a smooth line. Here, for example, are four successive magnifications of regions from a famous fractal, the Mandelbrot set. Starting from the upper left and moving clockwise, each step is a 10-fold magnification of the yellow box region in the previous image.

Mandelbrot Magnified 1Mandelbrot Magnified 2
Mandelbrot Magnified 4Mandelbrot Magnified 3

What makes this so interesting is that the complexity arises from simple mathematical expressions. For example, the images of the Mandelbrot set and Julia sets are generated using the function

zn+1 = f (zn) = zn2 + c (Equation 1)

Equation 1 produces a series of numbers (z0, z1, z2, z3,...). To get the next number in the series (zn+1), you take the number you have now (zn), square it, and a constant (c). You keep on doing this, plugging in your newly-generated number to get the next one (this is called "iterating" the function). All you need to get started are two values: z0 and c. For the Mandelbrot set, z0 varies and c is fixed; for Julia sets, z0 is fixed, and c varies.

Probably the hardest part of this whole project is this: the variable z in Equation 1 is a complex number, but don't let that scare you. If you understand how to graph an (x, y) value, you can understand complex numbers (see the Bibliography for more information). For now, just think of each zn as a pair of numbers (x, y).

So how does Equation 1 end up producing the amazingly complicated Mandelbrot set images? Here's how it works. Remember that Equation 1 produces a series of numbers, and that each series begins with a value for z0 (c is fixed at 0 for the Mandelbrot set). How each series behaves depends on what you pick for z0. For values far from zero, the series rapidly approaches infinity. For values closer to zero, the series can remain bounded. In fact, the Mandelbrot set is the set of all z0 values for which Equation 1 does remain bounded. To generate the images shown above, the computer program treated the rectangular viewing area as if it was a small piece of the plane. So each pixel in that region corresponds to an (x, y) coordinate. For each pixel, that (x, y) value is plugged in for z0 and Equation 1 is iterated. If the resulting series remains bounded, that pixel is colored black. If the resulting series heads off towards infinity, that pixel is assigned a color (or gray value, as in the examples above).

How does the computer program know if the series is heading off towards infinity? Each time the program iterates Equation 1, it checks to see if the resulting value is inside or outside of a circle of radius 2.0, with its center at the origin. If the value is inside this circle (called the "escape" region—you'll see why in a second), the program goes on to calculate the next value. If the value is outside the circle, it means the series is going to head off towards infinity (it's escaped!), so the program stops iterating Equation 1, colors the pixel in, and moves on to the next one. The color assigned to a pixel whose series has "escaped" depends on how many iterations (steps) it took before the series left the escape region. Every pixel whose series escaped in 1 step is assigned color 1; every pixel whose series escaped in 2 steps is assigned color 2, and so on. One last thing: the program mustn't keep iterating forever, so a limit is set on how many times to iterate for each pixel. If the series hasn't escaped before the limit is reached, the pixel is colored black. It's a simple procedure, but it produces some amazingly complicated images.

Here are some examples for four different starting points (circled in yellow). The white lines show the progression of points in the series. In the first row are points within the Mandelbrot set. For these particular points, an endless, repeating sequence resulted (the iterations end up back at the starting point). In the second row are points in the colored region of the image, thus we know that the corresponding series "escaped." The white lines show how many steps it took. For points further from the edge of the Mandelbrot set, it takes fewer iterations before the series "escapes" (see image at lower left). For points right on the edge of the set, it generally takes a large number of iterations before the series "escapes" (see image at lower right). You can explore this yourself with the Fractal Microscope program (see Bibliography), using its "Orbits" feature.

Orbits 1Orbits 2
Orbits 3Orbits 4

Do fractals have any uses besides generating intriguing images? As you might expect, there are applications in computer graphics. Fractals can be used to generate more realistic-looking objects from simple algorithms. Fractals have also been used in cell-phone antenna design (see Bibliography). One of the interesting things discoveries in mathematics (and a lot of scientific discoveries, for that matter) is that it can take awhile before someone realizes how to apply the new knowledge. Sometimes the ideas have to percolate through to completely different areas of science before someone realizes "Aha, this is just what I've been looking for to help me solve..." If you find something interesting, go ahead and study it. Do your background research to see how your ideas fit with what is already known. Do your work well, and publish your results. If you think it's interesting, chances are, someone else will find your results interesting, too. Your discoveries may even help someone solve a problem that you don't even know about.

Something that you can have fun exploring is the relationship between a point in the Mandelbrot set and the Julia set corresponding to that point. Remember that both sets come from the same equation, but with different starting values. For the Mandelbrot set, z0 varies and c is fixed, while the reverse is true for Julia sets. The Bibliography has several different programs you can use to explore how the sets are related, and the Experimental Procedure will help you get started.

Terms, Concepts and Questions to Start Background Research

To do a project on fractals, you should do research that enables you to understand the following terms and concepts:

  • fractal
  • iteration
  • complex number
  • Mandelbrot set
  • Julia sets

Bibliography

Materials and Equipment

  • computer with Internet access
  • Web browser
  • printer

Experimental Procedure

  1. First, do your background research and make sure that you understand how a computer program generates images of the Mandelbrot set. It will make it much easier for you to understand how to use the programs.
  2. Use the Fractal Microscope and the NLVM applet to explore both the Mandelbrot set and the Julia sets. Your background research should give you some ideas of things to try to look for relationships between points in the Mandelbrot set and the corresponding Julia sets.
  3. Read the instructions for each program. The NLVM applet is simpler, but both programs have controls to help you with your explorations. The instructions for the Fractal Microscope are very helpful. Keep thinking back to how the images are generated. With your understanding of how the algorithms work to make the images, combined with the information in the instructions, you should be able to understand each of the program controls.
  4. Take notes as you explore. The Fractal Microscope has a Parameters window that you can open so you can keep track of where you are in the image. Write down the locations of interesting images, so that you can recreate them later.
  5. You can also save images from the Fractal Microscope, and download them to your computer for printing. Collect images that show how Julia sets relate to different points in the Mandelbrot set. Make sure to keep track of the starting points.
  6. For your display, you can have an image of the Mandelbrot set, surrounded by sample Julia set images, with arrows showing their c value on the Mandelbrot set.
  7. Remember objective #3: you should be able to explain what you've learned in your own words. Practice by explaining your results to your parents or a friend.

Variations

  • Another way to generate fractal images is called Iterated Function Systems (IFS). Instead of studying Mandelbrot and Julia sets, investigate how IFS-type fractals are generated. The Sierpinski Triangle, Von Koch snowflake, and various fern-like or tree-like fractal shapes are all well-known examples.

Credits

Andrew Olson, Science Buddies

What Makes a Team's Winning Percentage Deviate from the Pythagorean Relationship?

Objective

The objective of this project is to determine why teams win many more or fewer games than they're expected to based on runs/goals/points scored and allowed.

Introduction

The Pythagorean relationship is a fundamental one in sports: it correctly predicts the records of 98% of all teams. But in 2% of cases, it fails. Why does it fail?

Terms, Concepts and Questions to Start Background Research

Pythagorean Relationship, Bill James

Experimental Procedure

Find teams that deviated substantially from their expected Pythagorean record (this information is available for baseball teams on www.baseball-reference.com). Then look at their situational statistics (this information is available on ESPN.com) and determine what they did that resulted in more or fewer wins than they otherwise should have had. Determine what the teams have in common.

Credits

Gabriel Desjardins

The Effects of Card Counting on a Simple Card Game

Objective

The objective of this project is to prove the best strategy for playing Hi-Lo using basic probability. Using computer simulations, you can verify that a particular strategy is correct and show what happens to the odds of winning when "counting cards."

Introduction

Photo

Hi-Lo is a very simple card game. A dealer ("the house") starts with a deck of cards and turns over the top card. The player then guesses whether the next card in the deck will be higher or lower than that card. This process of turning over a card and guessing high or low continues through the rest of the deck. The best strategy allows the player to guess correctly more than 70% of the time. Interestingly, a consequence of the Law of Large Numbers is that remembering which cards have come up already - ie "counting cards" - does not substantially increase a player's chances of guessing correctly. This game becomes more complex if the player can bet on his or her guesses. The player can use a simple strategy to increase his or her expected winnings by 50% compared to always betting the same amount. More significantly, the player can also "count cards" and pick up an even larger advantage.

Terms, Concepts and Questions to Start Background Research

Probability theory, card counting, the law of large numbers

Experimental Procedure

First, you should play the game a bit by yourself. Develop your own strategy for playing the game (for example, if the card is greater than 7 you always guess lower, if the card is less than 7 you always guess higher) and then test it out by taking a deck of cards and keeping track of how often you can correctly guess whether the next card is higher or lower than the one you turned over. If you can guess right more than 70% of the time, you've probably got the right strategy. The next thing you need to do is pick a programming language. If you've never programmed before, you should start with QBASIC, which is available for free at many internet sites, and is as close to English as any programming language. You'll write a computer simulation that will play thousands of hands of this card game Your simulation program needs two parts - the first is a shuffling routine to make sure the deck is random. The second is the actual game-playing strategy. You'll also have to write some data collection routines so you know how many times you've won or lost, and on which cards. Programming the strategy is the most involved part of this project, and can lead to a lot of results about how to play the game.

Variations

A more advanced project would examine different betting and card counting strategies to determine the optimal betting strategy in different circumstances. You can also determine how much "the house" should pay to a player who guesses correctly, how many decks "the house" should use to discourage card counting, and how far the dealer should deal into the decks before shuffling and starting over.

Throwing You Some Curves: Is Red or Blue Longer?

Objective

The objective of this project is to prove that the sum of the perimeters of the inscribed semicircles is equal to the perimeter of the outside semicircle.

Introduction

The figure below shows a semicircle (AE, in red) with a series of smaller semicircles (AB, BC, CD, DE, in blue) constructed inside it. As you can see, the sum of the diameters of the four smaller semicircles is equal to the diameter of the large semicircle. The area of the larger semicircle is clearly greater than the sum of the four smaller semicircles. What about the perimeter?

Your goal is to prove that the sum of the perimeters of the inscribed semicircles is equal to the perimeter of the outside semicircle.

Figure 1 (applet or image): Prove that the sum of the perimeters of the inscribed semicircles is equal to the perimeter of the outer semicircle.

Notes on How to Manipulate the Diagram

The diagram is illustrated using the Geometry Applet (by kind permission of the author, see Bibliography). If you have any questions about the applet, send us an email at: scibuddy@sciencebuddies.org. With the help of the applet, you can manipulate the figure by dragging points.

In order to take advantage of this applet, be sure that you have enabled Java on your browser. If you disable Java, or if your browser is not Java-capable, then the figure will still appear, but as a plain, still image.

If you click on a point in the figure, you can usually move it in some way. The free points, usually colored red, can be freely dragged about, and as they move, the rest of the diagram (except the other free points) will adjust appropriately. Sliding points, usually colored orange, can be dragged about like the free points, except their motion is limited to either a straight line, a circle, a plane, or a sphere, depending on the point. Other points can be dragged to translate the entire diagram. If a pivot point appears, usually colored green, then the diagram will be rotated and scaled around that pivot point. (Note that figures will often use only one or two of the above types of points.)

You can't drag a point off the diagram, but frequently parts of the diagram will be moved off as you drag other points around. If you type r or the space key while the cursor is over the diagram, then the diagram will be reset to its original configuration.

You can also lift the figure off the page into a separate window. When you type u or return the figure is moved to its own window. Typing d or return while the cursor is over the original window will return the diagram to the page. Note that you can resize the floating window to make the diagram larger.

The figure truly illustrates the fact that the position of the points along the line is entirely arbitrary: the proof will hold in any case.

Terms, Concepts and Questions to Start Background Research

To do this project, you should do research that enables you to understand the following terms and concepts:

  • radius of a circle,
  • diameter of a circle,
  • circumference of a circle,
  • π,
  • mathematical proof.

Bibliography

Materials and Equipment

  • For the proof, all you'll need is:
    • pencil,
    • paper,
    • compass, and
    • straightedge.
  • Here's a suggestion for your display: in addition to your background research and your proof, you can make a model of Figure 1 with colored paper. Use a compass and straightedge to construct the semicircles. Cut pieces of string or yarn equal to the arc-lengths of the semicircles. You can use these to demonstrate that the perimeter lengths are indeed equal.

Experimental Procedure

  1. Do your background research,
  2. organize your known facts, and
  3. spend some time thinking about the problem and you should be able to come up with the proof.

Variations

Credits

Andrew Olson, Science Buddies
Alexander Bogomolny, for the idea
Professor David Joyce, for the Geometry Applet

The Birthday Paradox

Objective

The objective of this project is to prove whether or not the birthday paradox holds true by looking at random groups of 23 or more people.

Introduction

Photo

The Birthday Paradox states that in a random gathering of 23 people, there is a 50% chance that two people will have the same birthday. Is this really true?

Terms, Concepts and Questions to Start Background Research

Birthday Paradox, probability theory, converse probability

Bibliography

There are a number of different sites that explain the Birthday Paradox and explain the statistics. Here is one to get you started:

http://en.wikipedia.org/wiki/Birthday_paradox

Experimental Procedure

1) First you will need to collect birth dates for random groups of 23 or more people. Ideally you would like to get 10-12 groups of 23 or more people so you have enough different groups to compare. Here are a couple of ways that you can find a number of randomly grouped people.

  • Most schools have around 25 students in a class, so ask a teacher from each grade at your school to pass a list around each of his/her classes to collect the birth dates for students in each of his/her classes.
  • Use the birth dates of players on major league baseball teams. (Note: this information can easily be found on the internet).

2) Next you will need to sort through all the birth dates you have collected and see if the Birthday Paradox holds true for the random groups of people you collected. How many of your groups have two or more people with the same birthday? Based on the birthday paradox, how many groups would you expect to find that have two people with the same birthday?

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