Probability Related Riddles

The Box: There is an empty box, and an unlimited supply of balls beside it. Each period of time, two fresh balls are put into the box, and one ball from the box is taken out and thrown away. The first time period is an hour, the second is 30 minutes, the third is 15 minutes, etc. After two hours, an infinite number of operations took place, and the process is terminated. Q: How many balls are in the box?
A: It depends. If for example the ball removed in each period is one of the two just put in to the box, then the answer is that an infinity of balls are in the box after two hours. However, if the ball removed is one of the balls which has been in the box for the longest period of time, then the answer is that the box is empty.
The question now is what happens if the ball removed is chosen randomly from the balls in the box?
The Envelopes: A prize show involves the participant choosing one envelope of money out of two. The two envelopes contain sums of money, one holding 10 times as much as the other one. After the participant chooses one envelope, it is opened the money in it is counted. Now the participant is always granted a grace period in which he/she may switch envelopes. One line of argumentation says: The participant has in hand $x. Since the participant had no way to know in advance which envelope contains more money, there is a 50% chance that the other envelope holds $10x and a 50% chance that it holds $x/10. This means that by switching the participant can increase his/her expected prize 5.5 times, therefore the participant should always switch.
A simple argument of symmetry suggests that the envelopes should be equally promising.
Q: What should the participant do?
The Strings: A bag contains a hundred strings. I randomly pick two string ends, tie them together and put them back in the bag. I repeat this until no string ends remain.
Q: What is the expectation of the number of loops in the bag?

To be continued.