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.
To choose the processor with no job we do: \({3}\choose{1}\)
With two processors left to do the work, the new possible assignments
(with just those two): \(2^n\)
Since the question states exactly one processor with no jobs we need to
minus the possible assignments from \(2^n\) where one of these two processors
also have no jobs: \(2^n-2\)
We end up with:
For n jobs, the probability that exactly one processor has no jobs is
\({3}\choose{1}\)(\(\frac{2^n-2}{3^n}\))
If n = 6:
n = 6
d6 = choose(3,1) * ((2^n-2)/3^n)
cat("The probability that exactly one processor out of three has no jobs out of six jobs:", d6*100, "%")## The probability that exactly one processor out of three has no jobs out of six jobs: 25.5144 %