Question

the venn diagram below shows the 15 students in ms. kellys class.
the diagram shows the memberships for the art club and the volleyball club.
note that \david\ 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 art club.\
let \\(b\\) denote the event \the student is in the volleyball club.\

(a) find the probabilities of the events below.
write each answer as a single fraction.
\\(p(a) = \\)
\\(p(b) = \\)
\\(p(a \text{ or } b) = \\)
\\(p(a \text{ and } b) = \\)
\\(p(a) + p(b) - p(a \text{ and } b) = \\)

(b) select the probability that is equal to \\(p(a) + p(b) - p(a \text{ and } b)\\).
\\(p(a)\\)
\\(p(b)\\)
\\(p(a \text{ or } b)\\)
\\(p(a \text{ and } b)\\)

Explanation:

Count elements in each region of the Venn diagram

$$
LATEXBLOCK0
$$

Calculate the required probabilities

$$
LATEXBLOCK1
$$

Identify the equivalent probability for part (b)

$$
LATEXBLOCK2
$$

Answer:

Question (a)

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

Question (b)

• \(P(A)\)
• \(P(B)\)
• \(P(A \text{ or } B)\) (Correct answer)
• \(P(A \text{ and } B)\)