WIN A CAR NOT GARBAGE

We are going to analyze the TV game show. 'Let us make a deal". There are three doors, A,B,and C. A car is kept behind one of the doors and Garbage is kept behind others. You are asked to choose one door behind which you think the car is present. Let us assume that you select the door 'A'. Then, the anchor of the show open one of the other two doors where garbage is present. Assume that he opens the door C. Now the anchor asks you whether you want to switch to door B or stick to door A, because the car may be present in door A or B. The C is already open. Here the statistics says, better to switch to door B from A. You will get more chance of winning the car". Let us find out how?

In the beginning of the show, there are three closed doors. If you choose one of them, you have 1/3 chance of winning the car. When one door is open and you are allowed to change your door, you get 2/3 chance of winning the car. If you stay with your original choice A, your chance also stays the same 1/3. If you switch over to another door B, your chance of winning increases to 2/3.(2 times 1/3=2/3)
If you toss a coin one time, the chance of winning a head is 1/2. If you loss it 2 times, the chance is 2*1/2 =1. One shows that there is certain win in two throws. If you throw 10 times, the chance is 10*1/2 =5. That is, you are likely to get head 5 times. As the number of tosses increases, the winning chance also increases. Similarly, if you go on choosing one door after another, your winning chance increases.

Let us look at the game show in another angle. Suppose the show is performed 300 times in a year. Each time the car is kept randomly behind one of the three doors. In all the 300 shows, the participants initially choose door A (assume) and switches to another door later. They are likely to win the car 200 times -2/3 chance. If they do not switch over, they will only win 100 times -1/3 chance. Here we are also assuming the game show is conducted honestly.

FOOT NOTE: The chance of winning is 1/6 means, out of 6 trials, you may win one time. Out of 600 trials, you many win 100 times. The moral of the story: Even a dull candidate will win if he tries many, many times.

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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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