Question

the venn diagram below shows the 12 students in ms. coxs 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 carlos 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)=

Explanation:

Step1: Calculate \(P(A)\)

The number of students in event \(A\) (Tennis Club) is 5 (Ahmad, Greg, Reuben, Leila, Charmaine). The total number of students is 12. So \(P(A)=\frac{5}{12}\).

Step2: Calculate \(P(B)\)

The number of students in event \(B\) (Soccer Club) is 7 (Lashonda, Josh, Tammy, Amy, Keith, Kala, Charmaine). So \(P(B)=\frac{7}{12}\).

Step3: Calculate \(P(A\cap B)\)

The number of students in both \(A\) and \(B\) (students in both Tennis and Soccer Clubs) is 2 (Leila, Charmaine). So \(P(A\cap B)=\frac{2}{12}=\frac{1}{6}\).

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

Use the formula \(P(B|A)=\frac{P(A\cap B)}{P(A)}\). Substitute \(P(A\cap B)=\frac{1}{6}\) and \(P(A)=\frac{5}{12}\) into the formula. \(P(B|A)=\frac{\frac{1}{6}}{\frac{5}{12}}=\frac{1}{6}\times\frac{12}{5}=\frac{2}{5}\).

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

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

Answer:

\(P(A)=\frac{5}{12}\), \(P(B)=\frac{7}{12}\), \(P(A\cap B)=\frac{1}{6}\), \(P(B|A)=\frac{2}{5}\), \(P(A)\cdot P(B|A)=\frac{1}{6}\)