An Old Probability Problem, Revisited

Alternatively, we can integrate along the y-axis for a more compact solution (see original article for notation).

An Old Probability Problem

This is an old challenge problem from my Data Management class with a very surprising answer.

What is a probability that a quadratic equation of the form x^2+2bx+c=0 (where b and c are real numbers) will have real roots?

Here is my solution:

I used the same technique stated here:

http://webs.wichita.edu/facsme/cbl/functions/roots.pdf

Remember that we are considering the probability of a point being in the space below the curve c=b^2 (condition for discriminant to be real). Therefore we consider the size of our square from -a to a on the horizontal axis and take this to the infinite limit (whole real line). We consider a one-sided problem due to the inherent symmetry of the curve b^2. The curve is integrated from 0 to sqrt(a) because this bounds the curve within a square of size a^2 in the first quadrant. The remainder area from sqrt(a) to a in the top quadrant is provided by the second term in the numerator. The last term is for the square in the bottom quadrant. The denominator simply doubles the square for top and bottom quadrants. Simplifying the limit, we obtain 1.

Caustics and Parametric Curves

Circle Catacaustic (Courtesy of Wolfram MathWorld)

If you have ever stared at a cylindrical glass of water sitting on table on a sunny day, you might have seen an intricate geometry of filaments of light projected onto the table behind the glass (in the direction away from the sun). This is called the caustic of the glass.

What exactly is a caustic? It is the envelope of rays from the reflection (catacaustic) or reflection (dicaustic) of light from an object. Actually the projected envelope from a glass is a nephroid (arising due to the differential equations involved) or a kidney-shaped object (see etymology).

It was this trivial experience, seeing something magical in a glass that started my obsession with optics/photonics. It also introduced me to the wonderful world of parametric curves, like the nephroid.

Parametric curves are interesting because they arise from very simple physical constraints (e.g. minimized degrees of freedom, path of least time, etc.). Consider the following problems:

Brachistochrone Problem
http://mathworld.wolfram.com/BrachistochroneProblem.html

Tautochrone Problem
http://mathworld.wolfram.com/TautochroneProblem.html

Other parametric curves such as involutes have important applications in engineering such as gears (so that a pair of gears mesh perfectly).
http://en.wikipedia.org/wiki/Involute_gear

Fun with Cryptography - Cyptoquip

Recently, I found a puzzle called Cryptoquip in my local newspaper, the Toronto Star.

The basic premise of this puzzle is a simple plaintext encoded using a substitution cipher, of which one letter clue is given (i.e. Today's Clue: L equals S). Using this clue and the properties of the message itself, you must solve the puzzle. It requires some logic, some intuition and also quite a bit of luck (so be patient).

Strategies:
Look for one-letter words (they have to be "A" or "I")
Look for common three-letter words (i.e. "THE")
Use hyphenated words, exclamations and quotations to your advantage (e.g. a quotation may be preceded by a word like "SAY")
Cross-check your guess letters in multiple locations, when possible.

With a little practice, you can become a skilled code-breaker. The punny messages are great fun to crack and I have to say that I am already addicted.

197 free Cryptoquips can be found here: http://www.cse.ucsd.edu/~mstepp/cryptoquip/crypto.html