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At East College, 7776 students are in the freshman class, 6750 are sophomores, 6750 are juniors, and the rest are seniors. About 18% of the students in each class are in the performing arts.

a. If there are 27000 undergraduates at the school, what is the probability of being a senior in the performing arts?

b. Is being in the performing arts independent of your class standing?

c. If a student is in the performing arts, what is the probability that he or she is a senior?

At East College, 7776 students are in the freshman class, 6750 are sophomores, 6750 are juniors, and the rest are seniors. Given that there are a total of 27000 undergraduates at the school, the number of seniors at East College can be calculated as follows:

[tex]\sf Freshman + Sophomores + Juniors + Seniors = 27000 \\\\7776 + 6750 + 6750 + Seniors = 27000 \\\\21276 + Seniors = 27000 \\\\Seniors = 27000 - 21276 \\\\Seniors = 5724[/tex]

Since about 18% of the students in each class are in the performing arts, the number of seniors who are in the performing arts is 18% of 5724:

[tex]\sf \textsf{Performing arts seniors} = 18\% \;of\; 5724 \\\\\textsf{Performing arts seniors} = 0.18 \times 5724 \\\\\textsf{Performing arts seniors} = 1030.32 \\\\\textsf{Performing arts seniors} \approx 1030[/tex]

So, approximately 1030 students are seniors in the performing arts.

The probability of a randomly chosen student being a senior in the performing arts is the number of seniors in the performing arts divided by the total number of students:

[tex]\sf \textsf{P(Senior and Performing Arts)} = \dfrac{1030}{27000} \\\\\textsf{P(Senior and Performing Arts)} = 0.03814814... \\\\\textsf{P(Senior and Performing Arts)} \approx 3.81\%[/tex]

Therefore, the probability of being a senior in the performing arts is approximately 3.81% (rounded to two decimal places).

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Part b

For independence, the proportion of students in the performing arts should be the same across all classes. Given that 18% of students in each class are in the performing arts, the percentage does not vary with class standing. This implies that the probability of being in the performing arts is the same regardless of class. Therefore, being in the performing arts is independent of class standing.

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Part c

To find the probability of being a senior given that a student is in the performing arts, we can use the conditional probability formula:

[tex]\sf P(\textsf{Senior} \mid \textsf{Performing Arts}) = \dfrac{P(\textsf{Senior and Performing Arts})}{P(\textsf{Performing Arts})}[/tex]

We know from our calculations in part a that the probability of being a senior in the performing arts is:

[tex]\sf P(\textsf{Senior and Performing Arts}) = \dfrac{1030}{27000}[/tex]

Since 18% of the students in each class are in the performing arts, the probability of a student being in the performing arts is 18%:

[tex]\sf P(\textsf{Performing Arts}) = 0.18[/tex]

Therefore:

[tex]\sf P(\textsf{Senior} \mid \textsf{Performing Arts}) = \dfrac{\frac{1030}{27000}}{0.18} \\\\\\ P(\textsf{Senior} \mid \textsf{Performing Arts}) = \dfrac{\dfrac{1030}{27000}}{0.18} \\\\\\ P(\textsf{Senior} \mid \textsf{Performing Arts}) =0.2119341563... \\\\\\ P(\textsf{Senior} \mid \textsf{Performing Arts}) \approx 21.19\%[/tex]

So, the probability that a student in the performing arts is a senior is approximately 21.19% (rounded to two decimal places).