THE BIRTHDAY PARADOX

If there are 60 people in a gathering, will any two persons have the same birth day? We are doubtful. we think if there are more than 365 persons, there may be a possibility of two persons sharing same birthday. But the statistics' answer is stunning. Here we go.

In a coin toss, chance of getting head or tail is 50 % or 1/2. Chance of head is 1/2 and the tail is 1/2. Hence the total probability is 1.

In a dice throw, chance of getting '1' is 1/6 and the chance of 'not' getting '1' is 5/6 = 1-1/6.

In a gathering, the chance for a person to have any date in a year as a birthday is 365/365=1=100%. For another person, the chance of not having a birthday same as first person is 364/365. For a third person, the chance of not having a birthday same as previous two persons is 363/365. Hence the probability for not having a match (two persons having same birth day ) among three persons is (365/365) * (364/365)*(363/365) (based on property of probability).

In another words, the probability of having a match among three persons is

1 - (365/365) * (364/365)*(363/365)

So the chance of finding a match of birthday among N persons is

= 1 - (365/365) * (364/365)*(363/365)* ........*(365-N+1/365)

= 1-[365*364*363*...........*(365-N+1)/365^N]

Using the above formula, let us calculate the probability of a match for different number of persons N = 1,2,3... and tabulate the results.

N P
no. of persons probability of a match

1 0
9 0.1 or 10%
23 0.5 or 50%
30 0.7 or 70%
58 1 or 100%

So the chance for two persons having same birthday (a match) is 50% even for 23 persons. A shocking result. If there are 58 persons in a room, there is a 100% chance for a match.Try to verify yourself.

This birthday problem(clash of dates, clash of data, clash of occupancy). Finds applications in spread of epidemics, photons occupancy and cryptography.

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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

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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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