$$ 5.5,5.6,5.7 $$ ) Puestion $$ 18,5.5 .61 $$ The grade appeal process at a university requires that a jury be structured by selecting four individuals randomly from a pool of five students and nine faculty. (a) What is the probability of selecting a jury of all students? (b) What is the probability of selecting a jury of all faculty? (c) What is the probability of selecting a jury of two students and two faculty? (a) What is the probability of selecting a jury of all students? 0.00500 (Round to five decimal places as needed.) (b) What is the probability of selecting a jury of all faculty? 0.12587 (Round to five decimal places as needed.) (c) What is the probability of selecting a jury of two students and two faculty? $$ \square $$ (Round to five decimal places as needed.)

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UpStudy AI · Accepted Answer

We first note that the total number of ways to select a jury of 4 from 14 individuals (5 students and 9 faculty) is given by

$$
\binom{14}{4} = \frac{14 \times 13 \times 12 \times 11}{4 \times 3 \times 2 \times 1} = 1001.
$$

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**(a) All Students**

We want to choose 4 students from 5. The number of ways to do this is

$$
\binom{5}{4} = 5.
$$

Thus, the probability of selecting an all-student jury is

$$
P(\text{all students}) = \frac{\binom{5}{4}}{\binom{14}{4}} = \frac{5}{1001} \approx 0.00500.
$$

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**(b) All Faculty**

We want to choose 4 faculty from 9. The number of ways to do this is

$$
\binom{9}{4} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126.
$$

Thus, the probability of selecting an all-faculty jury is

$$
P(\text{all faculty}) = \frac{\binom{9}{4}}{\binom{14}{4}} = \frac{126}{1001} \approx 0.12587.
$$

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**(c) Two Students and Two Faculty**

We choose 2 students from 5 and 2 faculty from 9. The number of ways is

$$
\binom{5}{2} \times \binom{9}{2}.
$$

Calculating each term:

$$
\binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10,
$$
$$
\binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36.
$$

Thus, the total number of ways is

$$
10 \times 36 = 360.
$$

So, the probability of selecting a jury of 2 students and 2 faculty is

$$
P(2 \text{ students and } 2 \text{ faculty}) = \frac{360}{1001} \approx 0.35964.
$$

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Final Answer for (c):

$$
\boxed{0.35964}
$$
The probability of selecting a jury of two students and two faculty is approximately 0.35964.

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