Question

the venn diagram below shows the 15 students in mr. andersons class. the diagram shows the memberships for the softball club and the art club. note that \leila\ is outside the circles since she is not a member of either club. one student from the class is randomly selected. let a denote the event \the student is in the softball club.\ let b denote the event \the student is in the art club.\ (a) find the probabilities of the events below. write each answer as a single fraction. p(a) = p(b) = p(a or b) = p(a and b) = p(a) + p(b) − p(a and b) = (b) select the probability that is equal to p(a) + p(b) − p(a and b). ∘ p(a or b) ∘ p(a and b) ∘ p(a)

Explanation:

Part (a)

Step 1: Find \( P(A) \)

• Explanation: Count students in Softball Club (A). Softball only: Jessica, Sam, Omar (3); Both: Donna, Milan, Lisa, Juan, Melissa (5). Total in A: \( 3 + 5 = 8 \). Total students: 15 (including Leila).
• \( P(A) = \frac{\text{Number in } A}{\text{Total}} = \frac{8}{15} \)

Step 2: Find \( P(B) \)

• Explanation: Count students in Art Club (B). Art only: Keiko, Reuben, Latoya, Isabel, Kareem, Henry (6); Both: 5. Total in B: \( 6 + 5 = 11 \).
• \( P(B) = \frac{\text{Number in } B}{\text{Total}} = \frac{11}{15} \)

Step 3: Find \( P(A \text{ or } B) \)

• Explanation: Students in A or B: Total - Leila (1). So \( 15 - 1 = 14 \).
• \( P(A \text{ or } B) = \frac{14}{15} \)

Step 4: Find \( P(A \text{ and } B) \)

• Explanation: Students in both clubs: Donna, Milan, Lisa, Juan, Melissa (5).
• \( P(A \text{ and } B) = \frac{5}{15} = \frac{1}{3} \)

Step 5: Find \( P(A) + P(B) - P(A \text{ and } B) \)

• Explanation: Substitute values: \( \frac{8}{15} + \frac{11}{15} - \frac{5}{15} = \frac{8 + 11 - 5}{15} = \frac{14}{15} \)
• \( P(A) + P(B) - P(A \text{ and } B) = \frac{14}{15} \)

Part (b)

By the principle of inclusion - exclusion for probability, \( P(A \text{ or } B)=P(A)+P(B)-P(A \text{ and } B) \). This is a fundamental formula in probability theory that accounts for double - counting the intersection when we simply add \( P(A) \) and \( P(B) \).

Answer:

\( \boldsymbol{P(A \text{ or } B)} \)

Final Answers for (a)

\( P(A)=\boldsymbol{\frac{8}{15}} \)

\( P(B)=\boldsymbol{\frac{11}{15}} \)

\( P(A \text{ or } B)=\boldsymbol{\frac{14}{15}} \)

\( P(A \text{ and } B)=\boldsymbol{\frac{1}{3}} \) (or \( \frac{5}{15} \))

\( P(A)+P(B)-P(A \text{ and } B)=\boldsymbol{\frac{14}{15}} \)

Final Answer for (b)

\( \boldsymbol{P(A \text{ or } B)} \)