I am trying to solve the Problem 51 from the book "Fifty Challenging Problems in Probability with Solutions"
Starting from an origin $O$, a particle has a 50-50 chance of moving 1 step north, or 1 step south, and also 50-50 chance of moving 1 step east and 1 step west. After the step is taken, the move is repeated from the new position and so on indefinitely. What is the chance that the particle returns to that origin?
In the solutions, the authors state that, if we call $P$ the probability of return to the origin and $Q=1-P$ the probability of no return, then the probability of exactly $x$ returns is
$$ P(x) = P^x Q\,. $$ This last equation is not clear to me. Why is there a factor of the probability of the no return $Q$? I understand that each time the particle crosses the origin, the problem starts over again. But why is he including also the case that the particle doesn't return? I thought that $P(x) = P^x Q$ is the probability of not returning to the origin in $x+1$ trials. I am confused...
here's the initial part of the solution of the author
You need to include Q and only one time since you come back x times but the last time you don't, which forms a whole process. If you don't include Q, then it means you come back infinitely.