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I am working on the following problem for my probability class and I am a little stuck:

A particle moves at each step two units to the right or one unit to the left, with corresponding probabilities $p$ and $q=1-p$. If the starting position is $z>0$ find the probability that the particle will ever reach the origin.

I know a lot of these problems are solved with a finite difference approach so I am trying that.

This is what I have done, let $f(k)=\text{Probability that we reach } 0 \text{ if we start at } k.$ Now if we start at $k$ then we have that in the next step we reach $k+2$ with probability $p$ or we reach $k-1$ with probability $q$.

So we have that: $f(k)=pf(k+2)+qf(k-1)$. At this point I don't really know what to do.

I appreciate any help.

Thanks!

User112358
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1 Answers1

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Usually solving any second order recurrence relation of the form $a_n= p a_{n-1} + q a_{n+2}$ requires finding the roots of the equation $$a^n= p a^{n-1} + q a^{n+2}\qquad \text{or,}\qquad a=p+qa^3$$ If the roots are $r_1,r_2$ and $r_3$ then $a_n = Ar_1^n+Br_2^n+Cr_3^n$

$A,B$ and $C$ are to be computed using the base cases $a_0,a_1$ and $a_2$

Shaswata
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