Question

the venn diagram below shows the 12 students in ms. jacksons 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 dante 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 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(a) p(a)·p(b|a) p(b|a)

Explanation:

Step1: Calculate total number of students

There are 12 students in total.

Step2: Calculate \(P(A)\)

There are 5 students in the Tennis - Club (Pablo, Rafael, Carlos, Raina, Ann). So \(P(A)=\frac{5}{12}\).

Step3: Calculate \(P(B)\)

There are 7 students in the Soccer - Club (Raina, Ann, Maria, Jane, Ivanna, Dan, Christine, Justin). So \(P(B)=\frac{7}{12}\).

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

The students in both clubs (intersection of \(A\) and \(B\)) are Raina and Ann. So \(P(A\ and\ B)=\frac{2}{12}=\frac{1}{6}\).

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

The formula for conditional probability is \(P(B|A)=\frac{P(A\ and\ B)}{P(A)}\). Since \(P(A\ and\ B)=\frac{2}{12}\) and \(P(A)=\frac{5}{12}\), then \(P(B|A)=\frac{\frac{2}{12}}{\frac{5}{12}}=\frac{2}{5}\).

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

\(P(A)=\frac{5}{12}\) and \(P(B|A)=\frac{2}{5}\), so \(P(A)\cdot P(B|A)=\frac{5}{12}\times\frac{2}{5}=\frac{1}{6}\).

Answer:

(a)
\(P(A)=\frac{5}{12}\)
\(P(B)=\frac{7}{12}\)
\(P(A\ and\ B)=\frac{1}{6}\)
\(P(B|A)=\frac{2}{5}\)
\(P(A)\cdot P(B|A)=\frac{1}{6}\)
(b) \(P(A)\cdot P(B|A)\)