FALSE POSITIVE AND DISEASE

In tossing coin, each event (toss) is independent event. The outcome of one toss does not influence the next toss. Hence the chance of getting a head is always 1/2.
Suppose, you left your car key in one of the four rooms of your apartment. Let us name the rooms as A,B,C and D. The probability of finding the key in the room A is 1/4. That is written as P(A) =1/4.
You searched the room A thoroughly and failed to find the key. Now the probability of finding the key in the room B is 1/3. It is written as P(B/A)=1/3. It is read as "probability of B given A". Remember the symbol.

A new disease has stroked the people. But only one (1%) of the people is affected by that disease. A Scientific lab has made a new 'device and kit' which detects the disease with 98% accuracy. But there is a catch. If a person does not have the disease, the device will falsely recognize the disease with 2% chance. It is a 'false positive'. 98% accuracy seems to be good for us. But let us see, how the story goes

P(A) - probability of population having disease = 1% = 0.01
P(B)- probability of population having No disease = 99% = 0.99
P(C/A) - probability of disease detected in affected population = 98%=0.98
P(C/B) - probability of disease detected (falsely) in normal population 2%=.02
we want:
P(A/C) = probability of affected population really detected by the device.

In statistics, Bayes theorem states that
P(A/C) = P(C/A)*P(A)/(P(C/A)*P(A) +P(C/B)*P(B))
P(A/C) = 0.98*0.01/(0.98*0.01+0.02*0.99)=33.1%

Hence the device's accuracy has gone down from 98% to 33%. What happened? The disease is so rare. If you test the general public , the number of people not having the disease will greatly out number the small group that has it. So even a very low rate of false positives will strongly impact the overall results.
Bayes' theorem is very intriguing statistical tool. It has many interesting and important applications.
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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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