A random walker moves in 2 dimension space. It starts from origin. $(x_0,y_0)=(0,0)$.
The random walker moves 4 directions with same probability. It means $(x_{n+1},y_{n+1})=(x_n+1,y_n)$ with probability $1 \over 4$. Same applies the move of $(x_n-1,y_n)$, $(x_n,y_n+1)$ and $(x_n,y_n-1)$
Let $P(n)\equiv$ probability of ($|x_n|<2|y_n|$ and $|y_n|<2|x_n|$ )
Then, $P(1)=0$,$P(3)=0$
$P(5)=p(|x|=2,|y|=3)+p(|x|=3,|y|=2)={80 \over 4^5} =0.078125$
$P(7)\approx 0.13$, $P(9)\approx0.17, P(11)\approx0.20$
Prove that $P(2k+1)>P(2k-1)$ for all $k\geq2$