Question

There are five faculty members in a certain academic department. These individuals have \(3,6,7,10\), and 14 years of teaching experience. Two of these individuals are randomly selected to serve on a personnel review committee. What is the probability that the chosen representatives have a total of at least 15 years of teaching experience? (Hint: Consider all possible committees.)

Short answer

The probability that the chosen representatives have a total of at least 15 years of teaching experience is 0.5 or 50%

Step-by-step solution

Identify All Possible Committees

The first step involves identifying all the possible combinations of 2 members committee out of 5 members. This is a 'combination' calculation since the order does not matter. The mathematical expression for combinations is usually given by: \(^nC_r = \frac{n!}{r!(n-r)!}\), where n is the total number of elements, r is the number of elements to choose, and '!' denotes factorial. So in this case, n=5 and r=2, hence the total combinations are given by \(^5C_2 = \frac{5!}{2!(5-2)!} = 10\). So, there are 10 possible committees.

Identify Committees With At Least 15 Years Experience

This step involves identifying all possible committees that have at least 15 years of teaching experience. Looking at the years of experience each faculty has, find all combinations that sum to at least 15. These are: (3, 14), (6,14), (6,10), (7,10), and (10,14). Therefore, there are 5 favourable committees.

Calculate the Probability

Probability of an event is given by the ratio of favorable outcomes to the total possible outcomes. Hence the probability (P) of selecting a committee with at least 15 years of experience would be given by \( P = \frac{total_favourable_committees}{total_possible_committees} = \frac{5}{10} = 0.5 \)