;5.7) This quiz: 12 point(s) possible $$ 1 \leftarrow $$ The grade appeal process at a university requires that a jury be structured by selecting four individuals randomly from a pool of eight students and eight 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? $$ \square $$ (Round to five decimal places as needed.) (b) What is the probability of selecting a jury of all faculty? $$ \square $$ (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

Let the total pool be $$8$$ students and $$8$$ faculty, so there are $$16$$ people. The total number of ways to choose a jury of $$4$$ is

$$
\binom{16}{4} = \frac{16 \times 15 \times 14 \times 13}{4 \times 3 \times 2 \times 1} = 1820.
$$

(a) For a jury of all students, we choose $$4$$ out of $$8$$:

$$
\binom{8}{4} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70.
$$

Thus, the probability is

$$
\frac{70}{1820} \approx 0.03846.
$$

(b) For a jury of all faculty, the calculation is the same (choosing $$4$$ out of $$8$$):

$$
\frac{70}{1820} \approx 0.03846.
$$

(c) For a jury of two students and two faculty, we choose $$2$$ students out of $$8$$ and $$2$$ faculty out of $$8$$:

$$
\binom{8}{2} \times \binom{8}{2} = 28 \times 28 = 784.
$$

Thus, the probability is

$$
\frac{784}{1820} \approx 0.43077.
$$

The answers rounded to five decimal places are:

(a) $$0.03846$$

(b) $$0.03846$$

(c) $$0.43077$$
(a) The probability of selecting a jury of all students is approximately 0.03846. (b) The probability of selecting a jury of all faculty is approximately 0.03846. (c) The probability of selecting a jury of two students and two faculty is approximately 0.43077.

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