Solution
= Solution
The first arrival time $T_1$, measured in hours after opening, has <exponential distribution> of rate twelve. Therefore
$$
\boxed{\mathbb P(1/4<T_1<1/2)=e^{-12/4}-e^{-12/2}=e^{-3}-e^{-6}.}
$$
Equivalently, there must be no arrivals in the first quarter hour and at least one in the next; <independent increments> give $e^{-3}(1-e^{-3})$.