Question

the venn diagram below shows the 14 students in ms. barness class.
the diagram shows the memberships for the tennis club and the soccer club.

a student from the class is randomly selected.
let \\(a\\) denote the event \the student is in the tennis club.\
let \\(b\\) denote the event \the student is in the soccer 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 diane is outside the circles since she is not a member of either club.

(a) find the probabilities of the events below.
write each answer as a single fraction.

\\(p(a) = \\)

\\(p(b) = \\)

\\(p(a \text{ and } b) = \\)

\\(p(b \mid a) = \\)

\\(p(a) \cdot p(b \mid a) = \\)

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

\\(\bigcirc p(b \mid a)\\)
\\(\bigcirc p(b)\\)
\\(\bigcirc p(a) \cdot p(b \mid a)\\)
\\(\bigcirc p(a)\\)

Explanation:

Count outcomes from the Venn diagram

$$
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Calculate probabilities for part (a)

$$
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Determine the equivalent probability for part (b)

$$
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Answer:

Question 1

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

• \(P(A) =\) <blank>\(\frac{9}{14}\)</blank>
• \(P(B) =\) <blank>\(\frac{1}{2}\)</blank>
• \(P(A \text{ and } B) =\) <blank>\(\frac{3}{14}\)</blank>
• \(P(B \mid A) =\) <blank>\(\frac{1}{3}\)</blank>
• \(P(A) \cdot P(B \mid A) =\) <blank>\(\frac{3}{14}\)</blank>

Question 2

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

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