Answer: 8/17
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Step-by-step explanation:
There are 7 professors and 9 instructors (first and last items mentioned).
So there are 7+9 = 16 people who are either a professor or an instructor, but not both. I'm assuming the job titles do not overlap, ie no one person has more than one title. The events "professor" and "instructor" are mutually exclusive. I'm leaving out the associate professors and assisant professors.
In total, there are 7+9+9+9 = 34 faculty members.
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We found there are 16 faculty members who are either a professor or an instructor, out of 34 total members.
This must mean the probability we want is 16/34 = 8/17
Side note: 8/17 = 0.4706 = 47.06% approximately