How come ?

1.

The Two Envelope Paradox

You have a choice between two envelopes that contain money. You are allowed to look at one before you chose. You are told that one envelope contains twice as much as the other. You pick one (let's say A). You find 10/=. So, the other envelope (B ) must have $5 or 20/=. Do you keep this one or go select B?
Well look at the expected value of the other one. It is (1/2 x 5/=) + (1/2 x 20/=) = 12.50/=. Hmm, looks like you need to pick B. But wait, what if you had picked it first? The same analysis would have caused you to conclude that A was the better one. How come?

2.

The Unexpected Execution (Hanging) Paradox

A prisoner is told that he will be hanged next week but the day of the hanging will be a surprise. The prisoner realizes that if he wakes up Saturday morning and finds himself not dead, then he can't be hanged that day because it would not be a surprise. By induction, he then eliminates Friday and so on for every day of the week. But come Wednesday he was hanged -- much to his surprise -- as the judge promised. How come its a surprise?


3.

The black card paradox.

Consider a stack of 7 playing cards, all of which are red except one which is black. It is your job to assemble the cards in a stack face down with the black one in some position. It is my job to turn the cards over one at a time until I get to the black one.

Can you arrange the cards in the deck in such a way that at every position, I will not be able to deduce that the next card is a black one before I turn it over? That is, as I go through the stack, one at a time, I will not be able to correctly deduce that the next card is black.

You cannot put it in the bottom, 7th, position, for I can certainly deduce that it is black if I get down to the last card and I haven't seen a black one. So that rules out the 7th position. There seems to be no doubt about that. (It would seem even that the 7th card is useless and we might as well play the game with 6, but I will let that pass.)

What about the 6th position? Well when I get down to the 6th card, I can deduce that the it must be black since we have already eliminated the 7th position. So you can't use the 6th position either.

Now, I say the 5th position has exactly the same problem. We have eliminated the 6th and 7th haven't we? This continues until we eliminate the 1st position.

This implies that i can always know it the card is black.

How come?

Magic with numbers

1. The Paradox of the digits

It's now official: 1=2

And here is the proof...

(1) Given X = Y
(2) XX = XY Multiply both sides by X
(3) XX - 2Y = XY - 2Y Subtract 2Y from both sides
(4) (X+Y)(X-Y) = Y(X-Y) Factor both sides
(5) (X+Y) = Y Cancel out common factors
(6) Y+Y = Y Substitute in from line (1)
(7) 2Y = Y Collect the Y's
(8) 2 = 1 Divide both sides by Y

Therefore 2 = 1 !!!!

Beware of the word INFINITY !!

1. The Infinite Circle

Nicholas of Cusa (1401-1464) made the following interesting point regarding the shape of an infinite circle. The curvature of a circle's circumference ecreases
as the size of the circle increases. For example, the curvature of the earth's surface is so negligible that it appears flat. The limit of decrease in curvature is a straight line.

An infinite circle is therefore... a straight line!!!!!


2. The Racetrack (or Dichotomy)

I can authoritatively say that one can never reach the end of a racecourse, for in order to do so one would first have to reach the halfway mark, then the halfway mark
of the remaining half, then the halfway mark of the final fourth, then of the final eighth, and so on ad infinitum.
Since this series of fractions is infinite, one can never hope to get through the entire length of the track (at least not in a finite time).

Start ____________________1/2__________3/4_____7/8__15/16... Finnish

But things get even worse than this...
Just as one cannot reach the end of the racecourse, one cannot even begin to run. For before one could reach the halfway point, one would have to reach the 1/4 mark, and before that the 1/8 mark, etc., etc. As there is no first point in this series, one can never really get started (this is known as the Reverse Dichotomy).

3. The Paradox of the Divided Stick

This is a big one..Ready??

This is a modern version of a plurality paradox asks what would happen if an infinitely divisible stick were cut in two, then half a minute later each half were again cut in two, then a quarter of a minute later each fourth cut in two, and so on ad infinitum.
At the end of one minute what would be left? An infinite number of pieces? Would each piece have any length?

From your answer lets continue...
This brings us to the two of the better-known plurality paradoxes :

(1) If something is divisible (the stick), then it is infinitely divisible (ie the stick can be divided into an infinite number of pieces). Now if each part has zero size, then the total has zero size, for an infinite number of zero lenghts add up to zero. If on the other hand each part(of the stick) has some finite size, then the total is infinite, for an infinite number of finite lenghts, however minuscule (small), must add up to an infinite total. So something divisible is either infinite or else has no size at all. Thus something finite is not divisible [the stick is not divisible]!!!!

(2) The total number of things is both finite and infinite. It is finite because, if there are many things, then there must be as many as there are "neither more nor less". And in that case their number is limited, hence finite. But on the other hand if there are many things, they must be infinite in number, for between any two there must always be others, and between those others still, and so on. (This paradox
apparently is meant to apply to spatial points, rather than to physical objects.)


4. Paradox of the chinkororo* [No pun intended]
This version of the Racetrack paradox brings out the conceptual difficulties inherent in infinite tasks. Suppose Paul Tergat wants to run the length of a racetrack but there are an infinite number of chinkororo (Tergat is not aware of them) who have the following intentions: the first chinkororo intends to paralyze Tergat if he reaches the halfway mark; the second intends to paralyze Tergat if he reaches the quarter mark; the third, if he reaches the one-eighth mark; and so on. As in the Reverse Dichotomy (last bit of paradox 4 above), Tergat cannot even start running: to do so would violate the intentions of an infinite number of chinkororo. However, it is not clear why he cannot start running, for until he does, no chinkororo has actually paralyzed him.

*If u r a kenyan you know and if not ask Nyachae....




Say something