Question

sixty high school students were surveyed about the type of books they prefer to read. the results are shown in the table below. what is the probability of a student preferring a mystery book, given that the student is a freshman?
freshmen mystery humor nonfiction
freshmen 5 6 3
sophomores 6 5 2
juniors 9 7 5
seniors 7 2 3
types of books preferred by high school students
options: \\(\frac{7}{30}\\), \\(\frac{5}{27}\\), \\(\frac{5}{14}\\), \\(\frac{1}{12}\\)

Explanation:

Step1: Find total freshmen

Sum freshmen's book preferences: \(5 + 6 + 3 = 14\)

Step2: Find mystery - preferring freshmen

From table, freshmen who like mystery: \(5\)

Step3: Calculate conditional probability

Use formula \(P(\text{Mystery}|\text{Freshman})=\frac{\text{Freshmen - Mystery}}{\text{Total Freshmen}}=\frac{5}{14}\)? Wait, no, wait: Wait, freshmen total is \(5 + 6 + 3 = 14\)? Wait, no, 5 (mystery) +6 (humor)+3 (nonfiction) = 14? Wait, but let's recalculate: 5+6=11, 11+3=14. Then the number of freshmen who like mystery is 5. So the probability is \(\frac{5}{14}\)? Wait, no, wait the options: one of the options is \(\frac{5}{14}\)? Wait the options are \(\frac{7}{30}\), \(\frac{5}{27}\), \(\frac{5}{14}\), \(\frac{1}{12}\). Wait, let's check again. Wait, freshmen: mystery 5, humor 6, nonfiction 3. So total freshmen: 5+6+3=14. Number of freshmen who like mystery: 5. So conditional probability \(P(\text{Mystery} | \text{Freshman})=\frac{\text{Number of freshmen who like mystery}}{\text{Total number of freshmen}}=\frac{5}{14}\). Wait, but let me check the table again. Freshmen row: Mystery 5, Humor 6, Nonfiction 3. So total freshmen: 5+6+3=14. So yes, 5/14. Wait but the option is \(\frac{5}{14}\)? Wait the third option is \(\frac{5}{14}\)? Wait the user's options: third one is \(\frac{5}{14}\). So that's the answer.

Wait, maybe I made a mistake. Wait, let's re - express:

Conditional probability formula: \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). In this case, \(A\) is "prefers mystery", \(B\) is "is a freshman". \(A\cap B\) is the number of freshmen who prefer mystery, which is 5. \(P(B)\) is the number of freshmen, which is \(5 + 6 + 3=14\). So \(P(A|B)=\frac{5}{14}\).

Answer:

\(\frac{5}{14}\) (corresponding to the option with \(\frac{5}{14}\))