Question

the venn diagram below shows the 12 students in ms. morriss class.
the diagram shows the memberships for the chess club and the science club.

a student from the class is randomly selected.
let \\(a\\) denote the event \the student is in the chess club.\
let \\(b\\) denote the event \the student is in the science 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 miguel 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) = \\)
\\(p(b) = \\)
\\(p(a \text{ and } b) = \\)
\\(p(a \mid b) = \\)
\\(p(b) \cdot p(a \mid b) = \\)

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

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 (a)

• \(P(A) = \frac{5}{12}\)
• \(P(B) = \frac{2}{3}\) (or \(\frac{8}{12}\))
• \(P(A \text{ and } B) = \frac{1}{6}\) (or \(\frac{2}{12}\))
• \(P(A \mid B) = \frac{1}{4}\) (or \(\frac{2}{8}\))
• \(P(B) \cdot P(A \mid B) = \frac{1}{6}\) (or \(\frac{2}{12}\))

Question (b)

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