SOA Exam P Question: Exponential Distribution Here is an Exam P problem as I have it.  That is, it was passed down to me from someone else and I am unsure if the wording is exactly as it was originally posted.  I've tried searching for this problem on various sites but cannot find a similar one.
Problem: A company insures two types of drivers: Basic and Preferred.  Basic claims come in at a rate that is exponentially distributed with a mean of 3.  Preferred claims come in at a rate that is exponentially distributed with a mean of 6.
What is the probability that the next preferred claim happens at least two days after the next basic claim given that a basic claim just happened?
Here's what I was thinking: Let $B$ be the number of days until the next basic claim happens and let $P$ be the number of days until the next preferred claim happens.  Thus $B$ ~ Exponential$\left(\frac13\right)$ and $P$ ~ Exponential$\left(\frac16\right)$.  We want to find Prob$\left(P\ge2+B|B=0\right)$.  This is where I keep getting stuck.  I'm sure this problem is relatively easy but I am not seeing it.  Any hints are greatly appreciated.
 A: Independence of the random variables that you call $P$ and $B$ is not explicitly stated in the problem, but needs to be assumed. 
We want $\Pr(P \ge 2+B)$. That is I think the natural interpretation of the phrase "the next preferred claim comes in at least two days after the next basic claim."  By the memorylessness of the exponential, the fact there has just been a basic claim does not affect the calculation. 
The joint density function  of $B$ and $P$ is $\frac{1}{18}e^{-x/3}e^{-y/6}$ in the first quadrant, and $0$ elsewhere. Call this $f(x,y)$. Note we are using the variable $x$ to refer to $B$ and the variable $y$ to refer to $P$. (I would probably have called $B$ by the name $X$, and $P$ by the name $Y$.)
There are no difficulties in integrating.
Now draw a picture identifying the part of the first quadrant where the condition $P \ge 2+B$ holds. That is the part of the first quadrant that is above the line $y=2+x$. After we have done that, writing down the appropriate integral is immediate. We get 
$$\Pr(P\ge 2+B)=\int_{x=0}^\infty \left(\int_{y=2+x}^\infty f(x,y)\,dy  \right)\,dx.$$
