The Geophysics building at the University of Northern California is scheduled to be seismically reinforced. The reinforcement will occur at a random time uniformly distributed in the next $3$ years. Suppose that during any fixed time interval of length $t$, the number of major earthquakes is Poisson with mean $\lambda t$. Find the probability that no major earthquake occurs of the Geophysics building.
In a Poisson Process the probability that no event occurs within a period of time $t$ is $P_0(t) = e^{-\lambda t}$. I'm tempted to say that because the probability density function of the reinforcement occurring is $\frac13$ I can multiply this function by $P_0(t)$ and integrate from $0$ to $3$. But I'm having trouble making this procedure rigorous.
Your plan will work. This is the law of total probability: for an event $A$ and a continuous random variable $X$, $$ \Pr[A] = \int \Pr[A\mid X=x] f_X(x)\,dx. $$ In this case, if we let $A$ be the probability of no earthquakes, and let $X$ be the number of years the building stays unreinforced, then $f_X(x)$ is $\frac13$ on $[0,3]$, and $\Pr[A \mid X=x] = e^{-\lambda x}$: the probability that in a time interval of length $x$, no events in the Poisson process will occur. The integral becomes $$ \Pr[A] = \int_0^3 \frac13e^{-\lambda x}\,dx. $$