Question

the venn - diagram below shows the 17 students in ms. daviss 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 felipe, michael, and maria are outside the circles since they are not members 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)

Count the number of students in Art Club. There are 8 students in Art Club. Total students are 17. So $P(A)=\frac{8}{17}$.

Step2: Calculate P(B)

Count the number of students in Dance Club. There are 7 students in Dance Club. So $P(B)=\frac{7}{17}$.

Step3: Calculate P(A and B)

Count the number of students in both clubs. There is 1 student (Rachel) in both clubs. So $P(A\text{ and }B)=\frac{1}{17}$.

Step4: Calculate P(B|A)

Use the formula $P(B|A)=\frac{P(A\text{ and }B)}{P(A)}$. Substitute $P(A\text{ and }B)=\frac{1}{17}$ and $P(A)=\frac{8}{17}$. Then $P(B|A)=\frac{\frac{1}{17}}{\frac{8}{17}}=\frac{1}{8}$.

Step5: Calculate P(A)·P(B|A)

Multiply $P(A)=\frac{8}{17}$ and $P(B|A)=\frac{1}{8}$. So $P(A)\cdot P(B|A)=\frac{8}{17}\times\frac{1}{8}=\frac{1}{17}$.

Answer:

$P(A)=\frac{8}{17}$
$P(B)=\frac{7}{17}$
$P(A\text{ and }B)=\frac{1}{17}$
$P(B|A)=\frac{1}{8}$
$P(A)\cdot P(B|A)=\frac{1}{17}$