Question

using a venn diagram to understand the multiplication rule for probability. the diagram shows the memberships for the tennis club and the soccer club for the 11 students in ms. coxs class. a student 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 trey 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 the Tennis Club (event \(A\)) is \(5\), and the total number of students is \(11\). So \(P(A)=\frac{5}{11}\).

Step2: Calculate \(P(B)\)

The number of students in the Soccer Club (event \(B\)) is \(7\), and the total number of students is \(11\). So \(P(B)=\frac{7}{11}\).

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

The number of students in both the Tennis and Soccer Clubs is \(3\), and the total number of students is \(11\). So \(P(A\ and\ B)=\frac{3}{11}\).

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

By the formula \(P(B|A)=\frac{P(A\ and\ B)}{P(A)}\), substituting \(P(A\ and\ B)=\frac{3}{11}\) and \(P(A)=\frac{5}{11}\), we get \(P(B|A)=\frac{\frac{3}{11}}{\frac{5}{11}}=\frac{3}{5}\).

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

\(P(A)\cdot P(B|A)=\frac{5}{11}\times\frac{3}{5}=\frac{3}{11}\).

Answer:

\(P(A)=\frac{5}{11}\)
\(P(B)=\frac{7}{11}\)
\(P(A\ and\ B)=\frac{3}{11}\)
\(P(B|A)=\frac{3}{5}\)
\(P(A)\cdot P(B|A)=\frac{3}{11}\)