Question

the venn diagram below shows the 14 students in ms. russell’s class. the diagram shows the memberships for the computer club and the soccer club. note that jim is outside the circles since he 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 computer club. let b denote the event the student is in the soccer club. (a) find the probabilities of the events below. write each answer as a single fraction. p(a) = \boxed{} p(b) = \boxed{} p(a and b) = \boxed{} p(a or b) = \boxed{} p(a) + p(b) - p(a and b) = \boxed{} (venn diagram: computer circle includes rafael, martina, frank, lashonda, reuben, kira, scott, kelisha; soccer circle includes amy, maya, greg, ralna, felipe, scott, kelisha; jim is outside both circles)

Explanation:

Step1: Count total students

Total students = 14 (including Jim).

Step2: Find \( P(A) \)

Event A: Student in Computer Club.
Computer Club members: Rafael, Martina, Frank, Lashonda, Reuben, Kira, Scott, Kelisha.
Number of Computer Club members = 8.
\( P(A) = \frac{\text{Number in A}}{\text{Total}} = \frac{8}{14} = \frac{4}{7} \).

Step3: Find \( P(B) \)

Event B: Student in Soccer Club.
Soccer Club members: Amy, Maya, Greg, Ralna, Felipe, Scott, Kelisha.
Number of Soccer Club members = 7.
\( P(B) = \frac{\text{Number in B}}{\text{Total}} = \frac{7}{14} = \frac{1}{2} \).

Step4: Find \( P(A \text{ and } B) \)

Students in both clubs: Scott, Kelisha.
Number in both = 2.
\( P(A \text{ and } B) = \frac{\text{Number in both}}{\text{Total}} = \frac{2}{14} = \frac{1}{7} \).

Step5: Find \( P(A \text{ or } B) \)

Using formula \( P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) \), or count directly:
Members in A or B: Rafael, Martina, Frank, Lashonda, Reuben, Kira, Scott, Kelisha, Amy, Maya, Greg, Ralna, Felipe.
Number = 13.
\( P(A \text{ or } B) = \frac{13}{14} \).

Step6: Verify \( P(A) + P(B) - P(A \text{ and } B) \)

\( \frac{4}{7} + \frac{1}{2} - \frac{1}{7} = \frac{8 + 7 - 2}{14} = \frac{13}{14} \), matches.

Answer:

s:
\( P(A) = \boldsymbol{\frac{4}{7}} \)
\( P(B) = \boldsymbol{\frac{1}{2}} \)
\( P(A \text{ and } B) = \boldsymbol{\frac{1}{7}} \)
\( P(A \text{ or } B) = \boldsymbol{\frac{13}{14}} \)
\( P(A) + P(B) - P(A \text{ and } B) = \boldsymbol{\frac{13}{14}} \)