Q:

Statistics: Bertrand's Box Paradox

Possible Duplicate:
Probability problem 

This is the Bertrand's Box Paradox I read about on Wikipedia:

Assume there is three boxes: 
a box containing two gold coins,
      a box with two silver coins
      and a box with one of each. 
After choosing a box at random and withdrawing one coin at random,
      if that happens to be a gold coin,
      the probability is actually 66% instead of 50%.
And the problem is equivalent to asking the question
      "What is the probability that I will pick a box with two coins of the same color?".

No matter how hard I try, I just couldn't comprehend this.. 
How is the possibility of picking a gold coin the same as the probability of picking a box with two coins of the same color?
Does this imply there is a 66% chance of picking a gold coin and a 66% chance of picking a sliver coin? 
If so, can we just say there is 50% chance of picking either one of them since both stand a 66% chance....?! and suddenly everything makes no sense..
[UPDATES] It is actually the probability of the remaining coin to be gold is 66% but not the probability of obtaining the gold coin is 66%.. I've misread it.... 
And everything makes sense now :D !

A:

The result is being incompletely quoted. Perform the experiment as described. Now suppose that you end up with a gold coin. The question is: what is the probability that the other coin in the same box is gold?
This probability is $\frac{2}{3}$.  Let's do an informal computation. It will be imprecise, but could be made precise by using the notion of conditional probability.
Imagine repeating the experiment $3000$ times. Then each box will be picked roughly $1000$ times. We will get a gold coin about $1500$ times. Out of these $1500$ times that we get a gold, it will have come from the two-gold box $1000$ times.
So if we restrict attention to the $1500$ times that we get a gold, about $1000$ of these times it will come from the two-gold box. So given that we got a gold coin, the probability the other coin in the same box is gold should be around $\frac{1000}{1500}$.

