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Chapter 6: Problem 97

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

Expert verified
The required probability is 0.6.

Step by step solution

01

Calculation of all possible combinations

The total number of faculty members is 5. So, to find out all the possible combinations of two members that can be selected, we should use the formula for combinations, which is \( C(n, r) = n! / r!(n - r)! \), where 'n' is the total number of elements and 'r' is the elements to choose. Substituting values, we get \( C(5,2) = 5! / 2!(5-2)! = 10. \) So, there are 10 possible combinations in which two members can be selected.
02

Calculation of favorable combinations

Next, we need to find out the combinations that satisfy the condition of at least 15 years of combined experience. To have at least 15 years of experience, several combinations of two member groups can be: (7,10), (6,10), (10,14), (3,14), (6,14), and (7,14). Counting these combinations, there are 6 of them that satisfy the requirement.
03

Calculation of Probability

The probability of an event happening is the ratio of the favorable outcomes to the total possible outcomes. Probabilities range from 0 to 1. From our previous steps, we know there are 10 total possible outcomes and 6 of them are favorable. Therefore, the required probability is \( P = \text{favorable outcomes} / \text{total outcomes} \), which, substituting the values, gives \( P = 6 / 10 = 0.6 \). Hence, the probability that the chosen representatives have a total of at least 15 years of teaching experience is 0.6.

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