I have the following question. (May be it is very simple, but I cannot find the answer).
Suppose I have a 1d random walk on integer numbers with equal pobabilities of unit step in either direction (i.e. from i to i+1 or i-1). My current position is i=X (suppose X > 0), which was reached from the origin i=0 after T steps.
QUESTION: What is the probability that out of previous P positions of the random walk, Q positions were on positive side (with i>0).
(Notice, that the order of sequence of Q positive positions among P previous positions does not matter and can be any).
Thanks in advance for any suggestions.