I have been recently given a problem on a standard brownian motion and the calculation of the conditional probability, but I am completely lost as where to start.
The problem follows: Let B(t) be standard Brownian motion. Evaluate: P(B(3)≤3|B(1)=1).
I am sure, this is a basic problem, but I just do not know where to start. I assume the probability would be: the join pdf of B(3)=3 and B(1)=1 divided by pdf of B(1)=1?
Thanks, T
Brownian motion is a martingale. So conditioning $B(1)=1$ is equivalent to assuming that you are starting a new Brownian motion, with initial value 1. So $P(B(3)\leq 3|B(1)=1)=P(B(2)\leq 2)$. Can you finish it from here?