Question

the venn diagram below shows the 14 students in ms. reeds 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 laura 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 and b)= p(b|a)= p(a)·p(b|a)= (b) select the probability that is equal to p(a and b). p(b) p(b|a) p(a) p(a)·p(b|a)

Explanation:

Step1: Count total students

There are 14 students in total.

Step2: Calculate \(P(A)\)

There are 6 students in the Art - Club. So \(P(A)=\frac{6}{14}=\frac{3}{7}\).

Step3: Calculate \(P(B)\)

There are 5 students in the Dance - Club. So \(P(B)=\frac{5}{14}\).

Step4: Calculate \(P(A\cap B)\)

There are 3 students in both clubs. So \(P(A\cap B)=\frac{3}{14}\).

Step5: Calculate \(P(B|A)\)

Using the formula \(P(B|A)=\frac{P(A\cap B)}{P(A)}\), substituting \(P(A\cap B)=\frac{3}{14}\) and \(P(A)=\frac{3}{7}\), we get \(P(B|A)=\frac{\frac{3}{14}}{\frac{3}{7}}=\frac{3}{14}\times\frac{7}{3}=\frac{1}{2}\).

Step6: Calculate \(P(A)\cdot P(B|A)\)

\(P(A)\cdot P(B|A)=\frac{3}{7}\times\frac{1}{2}=\frac{3}{14}\).

Answer:

\(P(A)=\frac{3}{7}\)
\(P(B)=\frac{5}{14}\)
\(P(A\ and\ B)=\frac{3}{14}\)
\(P(B|A)=\frac{1}{2}\)
\(P(A)\cdot P(B|A)=\frac{3}{14}\)
(b) \(P(A)\cdot P(B|A)\)