Question 59
A fat coin is one which, when tossed, has a $\frac{2}{5}$ probability of being heads, $\frac{2}{5}$ of being tails, and $\frac{1}{5}$ of landing on its edge. Mr. Fat starts at $0$ on the real line. Every minute, he tosses a fat coin. If it's heads, he moves left, decreasing his coordinate by $1$; if it's tails, he moves right, increasing his coordinate by $1$. If the coin lands on its edge, he moves back to $0$. If Mr. Fat does this ad infinitum, what fraction of his time will he spend at $0$?