# evaluate conditional probability in brownian motion

Let $$W_t$$ be a standard brownian motion, and let $$0 < x < y$$. We want to calculate:

$$\mathbb{P}(W_y > 0 \vert W_x > 0)$$.

I am pretty stuck on how to do this. The only insight I have is that we would probably want to take advantage of the fact that increments are independent in a Brownian motion to rewrite $$W_x, W_y$$ to evaluate this conditional probability, but I don't know how to do that or set up the integral that would probably result. I am new to this topic, so I apologize if this is a dumb question.

• One way to get started is to note that the pair $(W_x,W_y)$ is jointly normal, with zero means, and covariance matrix $\left[\matrix{x&x\cr x&y\cr}\right]$. – John Dawkins Apr 15 at 15:42
• @JohnDawkins hmm, that makes sense! Then I can find out what $P(W_x, W_y) > 0$ is. But I would also need to calculate what $P(W_x > 0)$ is to find the full expression, do you have a suggestion for that? Thank you very much! – 0k33 Apr 15 at 21:00
• But $W_X$ is mean zero normal, so $P(W_x>0)=1/2$. – John Dawkins Apr 15 at 21:55
• @JohnDawkins oh man. I was sleeping on that one. Thank you so much, I know what to do now! If you want to put your comments as an answer, I will accept it, but if not that's also ok :) – 0k33 Apr 15 at 22:27