MONTE CARLO TECHNIQUE

Imagine a square of unit side. Inscribe a quarter circle within the square. The area of the square is 1 unit. The area of the quarter circle is pi/4 unit.

Let us randomly select N points within the square. Out of N points, let C points fall within the circle. Logically, the area of the square is proportional to N points and the area of the circle is proportional to C points
.
Circle: pi/4 proportional C
Square: 1 proportional N.
Dividing,
( pi/4)/1= C/N
pi = 4*C/N

So we can estimate pi in this method. As the number of random points increases, the accuracy of the pi increases. A computer program is highly suitable to implement this technique. This method is named monte carlo technique because monte carlo is famous for casinos based on random numbers.

WHAT THE PROGRAM SHOULD DO?
For each point; up to N points
1. Select a point randomly within the square.
LOOP 2. Check if a point falls within the circle.
3. Count the point falling in the circle in C.
Next point
4.Find the ratio C/N and multiply by 4 to get pi.

APPLICATIONS
This is the versatile problem solving method. When all other methods fail, this "random method" may come to our rescue. It used in many fields. For example: numerical integration, solving system of equations, searching, area of irregular shapes etc,.

Science update: The diameter of the observable universe is known to be about 93 billion light years. To calculate the circumference of a circle with such diameter accurately only 39 decimal places of "pi" are needed.

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LISSAJOUS FIGURES

Definition: "When a particle is subjected to two sine wave motion or two oscillatory motion at right angles, the particle describes lissajous figures". We know sine wave motion and circular motion is basically same. Hence we draw two circles A and B perpendicular to each other. The circle B rotates twice faster than circle A. That is, frequency of circle B is two times than that of A. A particle at the intersection of two circles is subjected to two sine wave motion A and B at 90 degree simultaneously. The particle will describe figures depending on the frequency and phase of A and B . In our case, the ratio of frequency is 1:2 and the two waves are in phase. To draw lissajous figures : A moving point in both the circles are chosen. Here we should remember; during the time taken by the circle A to complete one rotation, circle B completes two. Hence the points are marked on the circles according to their speed. Then straight lines

THE PARABOLA

A jet of water shooting from a hose pipe will follow a parabolic path. What is the so special about parabola. Y= x^2 Draw a graph for the above equation. It will result in a parabola. This parabola is also called unit parabola. Any equation involving square will yield a parabola. Example: Y = 2x^2 +3x+3 (also called quadratic equation) X= 2 and -2, both satisfies the equation 4 = X^2. Parabolic equations always have two solutions. Any motion taking place freely under gravity follows parabolic path. Examples: An object dropped from a moving train, A bomb dropped from flying plane, A ball kicked upwards. If a beam of light rays fall on the parabolic shaped mirror, they will be reflected and brought to focus on a point. This fact is made use of in Dish Antenna, Telescope mirrors, etc. Inverted parabola shape is used in the construction of buildings and bridges. Because the shape is able to bear more weight. A plane

CASINO'S GAME

Let us find out how the casino survives with mathematics. Say, your friend invite you for a game of dice. You must bet (wager) 2 dollars. If you roll 'six' you will get back 8 dollars. The game will go on for 30 rounds. All sounds good. The probability of rolling 'six' is 1/6. Since the game will be played for 30 times, the 'expected win' is 30*1/6 = 5. That is, you are expected to win 5 rounds out of 30. Hence your gain will be 5 * 8 =40 dollars. ok. This also implies that you will loose 25 rounds. Hence your loss will be 25*2 =50 dollars. Your net gain will be gain-less = 40-50 = -10 dollars. For 30 rounds, the loss is -10 dollars, Hence, for one round =-10/30 = -1/3 dollars. There will be a loss of -1/3 or 0.33 dollars per round. It is not a fair game. Let us make a simple formula to calculate 'Pay out per round\. The probability for a win = p The pay-out in case of win = V No. of rounds = n The expect

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