Summary: Consider the first exit time of one-dimensional Brownian motion \(\{B_s\}_{s\ge 0}\) from a random passageway. We discuss a Brownian motion with two time-dependent random boundaries in quenched sense. Let \(\{W_s\}_{s\ge 0}\) be an other one-dimensional Brownian motion independent of \(\{B_s\}_{s\ge 0}\) and let \(\mathbb{P}(\cdot|W)\) represent the conditional probability depending on the realization of \(\{W_s\}_{s\ge 0}\). We show that \[ -t^{-1}\ln\mathbb{P}^x (\forall_{s\in[0,t]}a+\beta W_s\le B_s\le b+\beta W_s|W) \] converges to a finite positive constant \(\gamma(\beta)(b-a)^{-2}\) almost surely and in \(L^p (p\ge 1)\) if \(a<B_0=x<b\) and \(W_0=0\). When \(\beta=1\), \(a+b=2x\), it is equivalent to the random small ball probability problem in the sense of equidistribution, which has been investigated in S. Dereich and A. Lifshits [Ann. Probab. 33, No. 4, 1397–1421 (2005; Zbl 1078.60029)]. We also find some properties of the function \(\gamma(\beta)\). An important moment estimation has also been obtained, which can be applied to discuss the small deviation of random walk with random environment in time (see [Y. Lv, “Small deviation for random walk with random environment in time”, Preprint, arXiv:1803.08772]).