Question

the venn - diagram below shows the 13 students in ms. martins 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 deandre 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 number of students

There are 13 students in total.

Step2: Count number of students in event A

There are 5 students in the Art - Club (Rafael, Karen, Manuel, Lena, Joe), so $n(A)=5$. Then $P(A)=\frac{n(A)}{n(S)}=\frac{5}{13}$.

Step3: Count number of students in event B

There are 8 students in the Dance - Club (Jose, Rachel, Deon, Juan, Leila, Jessica, Salma, Manuel, Lena, Joe), so $n(B)=8$. Then $P(B)=\frac{n(B)}{n(S)}=\frac{8}{13}$.

Step4: Count number of students in $A\cap B$

There are 3 students in both the Art and Dance Clubs (Manuel, Lena, Joe), so $n(A\cap B)=3$. Then $P(A\ and\ B)=\frac{n(A\cap B)}{n(S)}=\frac{3}{13}$.

Step5: Use the formula for conditional probability

The formula for conditional probability is $P(A|B)=\frac{P(A\cap B)}{P(B)}$. Since $P(A\cap B)=\frac{3}{13}$ and $P(B)=\frac{8}{13}$, then $P(A|B)=\frac{\frac{3}{13}}{\frac{8}{13}}=\frac{3}{8}$.

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

$P(B)\cdot P(A|B)=\frac{8}{13}\times\frac{3}{8}=\frac{3}{13}$.

Answer:

$P(A)=\frac{5}{13}$
$P(B)=\frac{8}{13}$
$P(A\ and\ B)=\frac{3}{13}$
$P(A|B)=\frac{3}{8}$
$P(B)\cdot P(A|B)=\frac{3}{13}$