Question

the venn - diagram below shows the 11 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 frank 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(a|b)= p(b)·p(a|b)=

Explanation:

Step1: Count total students

There are 11 students in total.

Step2: Find \(P(A)\)

There are 5 students in the Tennis Club (\(A\)). So \(P(A)=\frac{5}{11}\).

Step3: Find \(P(B)\)

There are 7 students in the Soccer Club (\(B\)). So \(P(B)=\frac{7}{11}\).

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

There are 4 students in both clubs. So \(P(A\ and\ B)=\frac{4}{11}\).

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

By the formula \(P(A|B)=\frac{P(A\ and\ B)}{P(B)}\), substituting values we get \(P(A|B)=\frac{\frac{4}{11}}{\frac{7}{11}}=\frac{4}{7}\).

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

\(P(B)\cdot P(A|B)=\frac{7}{11}\times\frac{4}{7}=\frac{4}{11}\).

Answer:

\(P(A)=\frac{5}{11}\)
\(P(B)=\frac{7}{11}\)
\(P(A\ and\ B)=\frac{4}{11}\)
\(P(A|B)=\frac{4}{7}\)
\(P(B)\cdot P(A|B)=\frac{4}{11}\)