I came across problem which I cannot seem to solve when self studying probability. I was wondering if people can help me out.
There are $n$ planets in an orbit. So, there are $n$ planets in acircle. An alien starts on one of the planets. With probability $p$ hecan move to the right planet. With probability $1-p$ he can move tothe left planet. What is the expected number of trips needed in orderfor the alien to visit all $n$ planets at least once.
Attempt:
Let $t$ be the number of trips or moves. We want to find $E[t|n]$.The base cases are easy: $E[t|n=1]=1$, $E[t|n=2]=2$.$E[t|n=3]$ becomes more challenging and I can't seem to find something that generalizes.I know this can also be modelled as a markov matrix, but unsure how to find the expected number of moves to visit all planets. Even assuming $p=\frac{1}{2}$ I cannot seem to find a solution or a good approach.