Attempting to solve the following problem I am confused about what to use as the probability density function
Problem
The time that it takes to service a car is an exponential random variable with rate 1. If A.J. brings his car in at time 0 and M.J. brings her car in at time t, what is the probability that M. J.'s car is ready before A. J.'s car? (Assume that service times are independent and service begins upon arrival of the car.)
Exponential Random Variable
A continuous random variable whose probability density function is given, for some $\lambda > 0$, by
$ f(x) = \left\{ \begin{array}{l l} \lambda e^{-\lambda x} &\text{if } x \ge 0\\ 0 &\text{if } x < 0 \end{array} \right. $
is said to be an exponential random variable (or, more simply, is said to be exponentially distributed) with parameter $\lambda$.
How I'm approaching the problem
My thought is to let $\lambda = 1$, let $f( x, y) = \lambda^2 e^{-\lambda x} e^{-\lambda y}$, let $X$ denote A. J.'s car, let $Y$ denote M. J.'s car and solve for $P\{ X < Y\} = \int\limits_0^\infty \int\limits_t^y f( x, y) dx dy$ but immediately I am troubled with the though that this is not the proper joint distribution function?