  We establish the singularity with respect to Lebesgue measure as a function
of time of the conditional probability that the sum of two one-dimensional
Brownian motions will exit from the unit interval before time $t$, given the
trajectory of the second Brownian motion up to the same time. On the way of
doing so we show that if one solves the one-dimensional heat equation with zero
condition on a trajectory of a one-dimensional Brownian motion, which is the
lateral boundary, then for each moment of time with probability one the normal
derivative of the solution is zero, provided that the diffusion of the Brownian
motion is sufficiently large.
