CALCULATE YOUR WAITING IN A QUEUE

What is the probability of getting an odd number in a dice throw? There are three odd numbers 1,3 and 5. Hence the probability is 3/6=50%.

Draw a circle within a square

Drop pins randomly in the square. The pin should fall on its tip. What is the chance of pin falling within the circle?

Area of the square = 2r*2r= 4r^2.
Area of the circle = pi r^2.
Hence the chance of a pin falling in the circle = area of the circle/area of the square.
= pi*r^2/4*r^2
= pi/4 = 3.14/4 = 0.7854
=78.54%
Hence the pin mostly will fall within the circle.

Let us assume that you are waiting in a queue in the doctor's clinic for treatment. Let us further assume that the waiting time varies between 0 to 45 minutes and the treatment time varies between 0 to 30 minutes . What is your chance of getting out of the clinic in 30 minutes? Let us name the waiting time as X and the treatment time as Y, then plot a graph.

We get a rectangle. Your time X and Y may fall at any point in the rectangle. Now what is your chance of total time not exceeding 30 minutes?
That is, X+Y must be less than 30.
Draw a line for the equation X+Y = 30 in the same graph. Any point in the shaded region below the line satisfies your condition.Now what is the chance to meet our condition?

The chance = Area of the shaded triangle/ area of the rectangle.
=0.5*b*h/l*b
=0.5*30*30/30*45
= 450/1350 = 0.33333
= 33.33%

So the chance is 33% which is not much. You should be prepared to wait for a longer time.
Here the probability and geometry is combined. These kind of problems can be solved using geometry. So statistics gives measurement for our expectations.

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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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