Let $B(t)$ be a Brownian Motion and $$M(t) = max_{s:s \leq t} B(s)$$ and $$\tau_a = min_t{B(t) = a}$$
Then, $P(\tau_a < t) = P(B(t) - B(\tau_a) > 0 \: |\: \tau_a < t) + P(B(t) - B(\tau_a) < 0 \: |\: \tau_a < t)$
So this says that the probability that the first time we hit 'a' has already occurred, is equal to the probability that the distance we have moved since we hit 'a' is greater than 0 plus the probability that the distance we have moved since we hit 'a' is less than 0? I don't understand why this is true, could someone please help me with the intuition?