Final answer:
The question asks about the time students spend during office hours modeled with exponential distribution. Specific numerical answers cannot be provided without the exact rates, but the methodology involves using the properties of exponential distribution to calculate expectations and probabilities.
Step-by-step explanation:
The questions involve the exponential distribution, which models the time between events in a Poisson process. We are considering each student's time in a professor's office as such an event. Unfortunately, without the specific rates for Ron, Sue, and Ted being part of the question, only a general approach to solving these problems can be provided.
(a) Expected Time Until Only One Student Remains
Considering each student's time follows an exponential distribution with a specific mean, we'd calculate the expectation of the minimum of the three.
(b) Probability Each Student is the Last to Leave
For each student, the probability that they are the last to leave would involve using the properties of exponential distributions comparing their rates to the sum of the rates of all students.
(c) Expected Time Until All Students are Gone
To find the expected time until all students are gone, we'd sum the expectations of each student's time, assuming independence between the lengths of their stays.
Note that without the specific rates, numerical answers cannot be given. Only the methodology to find the solution has been presented.