Some old wine...

Well, take this old probability problem.
A mathematician carries two matchboxes(in order to complicate life!), both initially containing 'n' sticks each.Whenever a stick is needed, he takes it from one of the boxes(boxes are distinguishable)but never keeps count of matches.Once it so happens that he takes out one of the box and finds it empty, takes out the other and that is also empty.What is the probability this happening?

A probability problem [S]

Let there be a biased coin with probability 'p' of getting head and probability 'q' of getting tail.We keep tossing the coins till we get first tail(head) after a series of non zero heads(tails).

(i)Calculate the average number of tossing required. [S]
(ii)Does this function take minima when p=q=1/2. [S]

The train Problem[R]

You and your friend are in two adjacent boggies of a train with infinite number of boggies.Both of you donot know in which boggie the other person is.Both of you move towards front or back boggie with the intention of meeting the other.A move consists of a person moving from one boggie to other and assume that both make moves simultaneously.What is the average(expected) number of moves after which both of you meet ?[reviewing the solution...]

Bug split

A bug either splits into two perfect copies of itself or dies. If the probability of splitting is p (and is independent of the bug's ancestry), what is the probability that a bug's descendants die out?