Question

the venn diagram below shows the 11 students in ms. kims class.
the diagram shows the memberships for the art club and the dance club.

a 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 dance club.\

the outcomes for the event \\(a\\) are listed in the circle on the left.
the outcomes for the event \\(b\\) are listed in the circle on the right.

note that lamar is outside the circles since he is not a member of either club.

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

(b) select the probability that is equal to \\(p(a \text{ and } b)\\).
\\(\bigcirc p(a \mid b)\\)
\\(\bigcirc p(b) \cdot p(a \mid b)\\)
\\(\bigcirc p(b)\\)
\\(\bigcirc p(a)\\)

Explanation:

Count the outcomes in each region of the Venn diagram

$$
LATEXBLOCK0
$$

Calculate the probabilities for part (a)

$$
LATEXBLOCK1
$$

Identify the equivalent probability for part (b)

$$
LATEXBLOCK2
$$

Answer:

Question 1

(a) Find the probabilities of the events below. Write each answer as a single fraction.

• \(P(A) =\) <blank>\(\frac{4}{11}\)</blank>
• \(P(B) =\) <blank>\(\frac{9}{11}\)</blank>
• \(P(A \text{ and } B) =\) <blank>\(\frac{3}{11}\)</blank>
• \(P(A \mid B) =\) <blank>\(\frac{1}{3}\)</blank>
• \(P(B) \cdot P(A \mid B) =\) <blank>\(\frac{3}{11}\)</blank>

Question 2

(b) Select the probability that is equal to \(P(A \text{ and } B)\).

• \(P(A \mid B)\)
• \(P(B) \cdot P(A \mid B)\) (Correct answer)
• \(P(B)\)
• \(P(A)\)