  In this paper we find an upper bound for the probability that a $3$
dimensional simple random walk covers each point in a nearest neighbor path
connecting 0 and the boundary of an $L_1$ ball of radius $N$. For $d\ge 4$, it
has been shown in [5] that such probability decays exponentially with respect
to $N$. For $d=3$, however, the same technique does not apply, and in this
paper we obtain a slightly weaker upper bound: $\forall \varepsilon>0,\exists
c_\varepsilon>0,$ $$P\left({\rm Trace}(\mathcal{P})\subseteq {\rm
Trace}\big(\{X_n\}_{n=0}^\infty\big) \right)\le \exp\left(-c_\varepsilon
N\log^{-(1+\varepsilon)}(N)\right).$$
