An animal leaves its burrow near a river bank and moves up and down along the river bank foraging for food. Its distance upstream from its burrow after t minutes is denoted by X(t), and may be reasonably modeled as an ordinary Brownian motion {X(t); t ≥ 0} with diffusion coefficient σ2 = 3 (meters)2 per minute.
a) if the animal is observed to be 6 meters upstream from its burrow after 5 minutes, find the probability that the animal will be less than 2 meters upstream from the burrow after 15 minutes.
b) If the animal is observed to be 30 meters upstream from its burrow after an hour, find the probability that it was downstream from its burrow after 20 minutes