[NW] Selecting a Jury
The grade appeal process at a university requires that a jury be structured by selecting five individuals randomly from a pool of eight students and ten faculty.


b. What is the probability of selecting a jury of all faculty?

Verified step by step guidance

1. Identify the total number of individuals in the pool. There are 8 students and 10 faculty members, so the total is \$8 + 10 = 18$ individuals.
2. Determine the total number of ways to select a jury of 5 individuals from the 18 people. This is a combination problem, so use the formula for combinations: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). Here, calculate \(\binom{18}{5}\).
3. Determine the number of ways to select a jury of 5 individuals all from the faculty. Since there are 10 faculty members, calculate \(\binom{10}{5}\).
4. Calculate the probability of selecting a jury of all faculty by dividing the number of favorable outcomes (all faculty juries) by the total number of possible juries. The probability is \(\frac{\binom{10}{5}}{\binom{18}{5}}\).
5. Interpret the result as the likelihood that a randomly selected jury of 5 people from the pool consists entirely of faculty members.

Key Concepts

Combinations

Combinations refer to the selection of items from a larger set where order does not matter. In this problem, we use combinations to calculate the number of ways to select jury members from students and faculty without regard to order.

Probability of an Event

Probability measures the likelihood of an event occurring, calculated as the ratio of favorable outcomes to total possible outcomes. Here, the event is selecting all faculty members for the jury.

Sample Space and Favorable Outcomes

The sample space is the total number of ways to select the jury from all candidates, while favorable outcomes are the selections meeting the event criteria (all faculty). Understanding these helps compute the probability accurately.