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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.

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    $\begingroup$ There seems to be a reflection principle lurking here. If you condition on the number of returns to 0 before hitting $i$, then reflecting each segment of returns lowers or raises $Q$ accordingly. It also might be useful to set up a recurrence of the form $P(p,q,i)$. $\endgroup$
    – Alex R.
    Apr 16, 2014 at 2:11

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