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

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.

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

• The Math Forum at Drexel University has some good advice on how to build a mathematical proof:
  http://mathforum.org/library/drmath/view/54693.html
• There are many more examples in their FAQ section:
  http://mathforum.org/dr.math/faq/faq.proof.html
• When you are constructing your proof, we sincerely hope that you will not have to resort to the following methods [grin]:
  http://www.bluemoon.net/~watson/proof.htm
• The Geometry Applet was kindly provided to Science Buddies by its author, Professor David Joyce, who wrote it for illustrating an amazing online edition of Euclid's Elements: http://aleph0.clarku.edu/~djoyce/java/elements/elements.html

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

• For a more advanced project, see
  Thinking in (Semi-)Circles: The Area of the Arbelos
• If you'd like to try making your own dynamic diagram with the Geometry Applet, check out these links:
  • Geometry Applet How-to Pages
  • Combining Computer Science and Math: Inscribing a Circle in a Triangle Using the Geometry Applet

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

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?

CAT 2007 EXAMINATION DATES OUT

CAT 2007 - Common Admission Test 2007 Notification, Dates

Common Admission Test (CAT 2007) for MBA Admissions in 2008 for in the following institutes will be held on Sunday, November 18, 2007, for their various programmes. :

• The Indian Institutes of Management (IIM ),Ahmedabad (IIM A)
• The Indian Institutes of Management (IIM), Bangalore (IIM B)
• The Indian Institutes of Management (IIM) Calcutta (IIM C)
• The Indian Institutes of Management (IIM) Indore (IIM Indore)
• The Indian Institutes of Management (IIM) Kozhikode (IIM K)
• The Indian Institutes of Management (IIM) Lucknow (IIM L)
• and other Management Institutes in India who accept CAT Scores .

IMPORTANT DATES TO CHECK:

CAT 2007 Important Dates
CAT Eligibility
CAT Bullien Download
IIM Admission Process
CAT 2007 Test Centers
CAT Test Pattern
HOW WAS CAT 2006 ?- Check out CAT 2006 Review Articles