WHY ONE GAME IS NOT ENOUGH TO DECIDE THE WINNER

Say, 'A' has so far won 60% of the games (chess, shuttlecock) he participated. But 'B' has won only 40% of the games. If A and B clash in a game, clearly A has 20% winning edge over 'B'. But , if luck favors B, A may lose. So one game is not fair enough to decide the winner.

Suppose A and B plays a set of three games, whoever wins any two games out of three is a clear winner. Let us see the maths behind it.

What is the chance for A to win any 2 games out of 3. There are four possibilities or pattern to win.

Pattern1. Win, win, win
Pattern2: win, win, lose
Pattern3. Lose,win,win
Pattern4. win, lose, win

The winning probability is 0.60 and losing probability is 0.40 for A. Hence substitute the probabilities and use multiplication rule of statistics.

Pattern1. .60*.60*.60 = 0.216
Pattern2. .60*.60*.40 =0.144
Pattern3 .40*.60*.60 = 0.144
Pattern4. .60*.40*.60= 0.144
Total winning probability = 0.648

In a set of three games, the winning chance of A is .648 or 64.8% and the remaining 100-64.8 = 35.2% is the winning chance of B. Now A has nearly 30% more edge over B.

So a set of three or five games will bring out the genuine winner.

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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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