A Really Cool Math Website!

[quantyst]

Give yourself a well-deserved break. Visit the following math website and savor it!

Amazing and all fascinating! =D>

http://www.mathematik.com/index.html

 

[ramnik]

Oh when it comes to cool websites on mathematics I like the following:

1. For puzzles easily the best is the UCB forum wu :: forums - Index

2. For nice(read tough/hardcore) mathematics, visit the problem solving groups page at NU Math Problem Solving Group Web Page

3. And then arXiv beats all when it comes to latest developments in mathematics.

These are the ones I can remember off hand! I'm sure there are plenty of awesome sites there. Choose as per taste and level of mathematics.

 

[sugi]

PETERSBURG PARADOX
Hi, I went into the website and found the interesting Petersburg Paradox.
The story is:
You entered in the Petersburg casino. Your entrance fee is 20 dollars in each game. In one game, a coin is thrown repetively until it shows no more head. You get 2n-20 Dollars, if n times Head appeared, the bank gets in each game 20-2n dollars.
You will experience that with this entrance fee, you are losing money. In reality, it is a mathematical fact that you are expected to win: you win with probability 2-n the amount of 2n Dollars. Summing over all n gives an infinite expectation value for your win. Whatever entrance fee you would pay, you would win. The paradox is that this disagrees with your experience to lose.

I go through to find out the expectation.
You will find that to get the breakeven, you need 1 trillion (10^12) trials.
In the mathematical world, it is true that it will end up with positive infinite numbers, but in this risky and limited world, the player will always lose.:D

PS: Sometimes, the more you try, the more you gain, and sometimes, the more you try, the more you lose. Wise man will understand his SWOT to understand the limitation.