3.15 A computing center has 3 processors that receive n jobs, with the jobs assigned to the processors purely at random so that all of the \(3^n\) possible assignments are equally likely. Find the probability that exactly one processor has no jobs.
First, as the question stem states, there is a total of \(3^n\) possible assignments. We are interested in finding a combination (since order does not matter) where one process is given 0 jobs. Either processor 1 or 2 or 3 could be assigned 0 jobs. Therefore, this essentially can be calculated with \({3 \choose 1}\). (Reference: https://www.mathsisfun.com/combinatorics/combinations-permutations.html)
As a result, the rest of the jobs are split into a combination with the other two computer processors. With only two processors left and n jobs, that leaves us \(2^n\) combination of jobs split between them. However, we need to ensure that computer processor (say A) and computer process (say B) each has at least one job. Therefore, we must exclude the combination with processor A having 0 jobs and B having n jobs, and processor B having 0 jobs and A having N jobs. Therefore, in reality, there are a total of \(2^n - 2\) combinations.
So to get the total combinations that exactly one processor has no jobs, we multiple \({3 \choose 1}\) (for one processor with zero jobs) with \(2^n - 2\) combinations to get the total of possible combinations. However, given that the question is asking for the probability, we need to divide this over the \(3^n\).
So the answer to this question is:
\[{3 \choose 1} (2^n - 2) / 3^n\]