A point of the lattice $\mathbb{Z}^3$ in $\mathbb{R}^3$ is painted white if at least one of its coordinates is odd. An ant is moving in $\mathbb{R}^3$.
At each integer time $t$ the ant is at a point in $\mathbb{Z}^3$ and it chooses one of points in $\mathbb{Z}^3$ at distance $1$ with uniform probability, and it moves there before time $t + 1$.
For an integer $n$, denote by $P_n$ the probability that among the previous $n$ integer times the ant was at least $90\%$ of the time at a white point.
Prove that $P_n$ decreases exponentially with $n$. Can you compute the rate?