SIMPSON'S PARADOX

Let us start with an example. Consider math and biology teachers working in a school. Every year, they train students and send to school final examinations. Some students pass. Some students do not make it. Every year, each teacher gets 'percentage of passes' score.
The pass percentages for each teacher for two years are given below.

Math teacher Biology teacher
Year I 70/80 = 87.5% 20/20 = 100%
Year II 10/20 = 50% 50/80 = 62.5%
Total 80/100 = 80% 70/100 = 70%

Here, the denominator is the number of students appeared and the numerator is the no.of students passed. Every year biology teacher seems to show good performance than the math teacher. But when we look at the total of 2 years, math teacher wins. This is Simpson's paradox.

Simpson's paradox occurs when groups of data show one particular trend, but this trend is reversed when the groups are combined together.

The paradox occurs when group sizes (no. of students, here) are different. Look at the column's of the table.

This Simpson's paradox should always be taken into account to avoid misinterpretation of statistical data.

For example, heart surgery and cataract surgery cannot be compared. Many people go for cataract surgeries and the success rate is high. Where as a few people opt for heart surgeries.

We always get complaints that women are not treated fairly in admissions, appointments etc. But, in some cases, women apply in less numbers, mostly they get opportunities. Hence, women group size and men group size are different and may lead to Simpson's paradox. Actually, an american university is sued for the gender gap arising out of Simpson's paradox.
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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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