Question

at a local college, 100 students were asked about their major fields of study. the results are shown below. what is the probability that a randomly selected student is a female or a liberal arts major?
majors at local college

	information technology	liberal arts	business, medicine
female	15	18	17

options: \\(\frac{7}{10}\\), \\(\frac{9}{16}\\), \\(\frac{9}{50}\\), \\(\frac{27}{50}\\)

Explanation:

Step1: Find total female students

Female in Information Science: 15, Liberal Arts: 18, Business (Assuming last column is Business? Wait, table has Information Science, Liberal Arts, and another. Wait, female row: 15 (Info), 18 (Liberal Arts), 17 (other). So total female: \(15 + 18 + 17 = 50\).

Step2: Find total Liberal Arts students

Male in Liberal Arts: 20, Female in Liberal Arts: 18. So total Liberal Arts: \(20 + 18 = 38\).

Step3: Find female and Liberal Arts overlap (female Liberal Arts: 18)

Using principle of inclusion - exclusion: \(P(\text{Female or Liberal Arts}) = P(\text{Female}) + P(\text{Liberal Arts}) - P(\text{Female and Liberal Arts})\)
Total students: 100.
\(P(\text{Female})=\frac{50}{100}\), \(P(\text{Liberal Arts})=\frac{38}{100}\), \(P(\text{Female and Liberal Arts})=\frac{18}{100}\)
So \(P=\frac{50 + 38 - 18}{100}=\frac{70}{100}=\frac{7}{10}\)? Wait no, wait female total: 15 + 18 + 17 = 50. Liberal Arts total: 20 + 18 = 38. Overlap: 18. So 50 + 38 - 18 = 70. 70/100 = 7/10? Wait but let's check again. Wait the table:

Male: Info 16, Liberal Arts 20, Other 14. Total male: 16+20+14=50. Female: Info 15, Liberal Arts 18, Other 17. Total female: 15+18+17=50. Total students: 50+50=100.

Liberal Arts: 20 (male) + 18 (female) = 38.

Female or Liberal Arts: number of female + number of Liberal Arts - number of female Liberal Arts.

Number of female: 50. Number of Liberal Arts: 38. Number of female Liberal Arts: 18. So 50 + 38 - 18 = 70. So probability is 70/100 = 7/10. Wait but let's check the options. First option is 7/10. Wait but maybe I made a mistake. Wait another way: count number of students who are female or Liberal Arts.

Female students: 15 (Info F) + 18 (LA F) + 17 (Other F) = 50.

Liberal Arts students: 20 (LA M) + 18 (LA F) = 38. But LA F is already in female, so total unique: 50 (female) + 20 (male LA) = 70. So 70/100 = 7/10. So the first option is 7/10.

Wait but let's recalculate:

Female: 15 + 18 + 17 = 50.

Liberal Arts: 20 + 18 = 38.

Female and Liberal Arts: 18.

So 50 + 38 - 18 = 70. 70/100 = 7/10. So the answer is 7/10.

Answer:

\(\frac{7}{10}\) (First option: \(\frac{7}{10}\))