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A bug starts at time $0$ at position $0$. At each step, the bug either moves to the right by $1$ step $(+1)$ with probability $1/2$, or returns to the origin with probability $1/2$. What is the expected number of steps for this bug to reach position $N$?

I tried to first find the possibility that this bug reaches $N$ as the number of steps goes to infinity. The recurrence equation I find is $$p_n = \frac{1}{2}p_{n-1}$$, where $p_n$ is the possibility for the bug starting at position $n$ to reach $N$. We also have the boundary condition $p_N = 1$. Then we see that $p_{N-1}=2$, and that $p_0 = 2^N$, which doesn't make sense at all because it is greater than $1$. I think I should sort out the value of probability first, and think about the number of expected steps later.

I'm sure there is something wrong with the recurrence equation, but what's wrong about it?

Lee
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vicissitude1999
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  • I think this is a famous one, but with a different concept. Wait for help. – manooooh Jan 15 '19 at 02:16
  • Are you asking about expected number of steps or probability? There is nothing wrong about expected number being greater than $1$. – David Jan 15 '19 at 02:33

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