Question

the venn diagram below shows the 11 students in mr. coxs class. the diagram shows the memberships for the music club and the tennis club. note that \henry\ and \debra\ are outside the circles since they are not members of either club. one student from the class is randomly selected. let a denote the event \the student is in the music club.\ let b denote the event \the student is in the tennis club.\ (a) find the probabilities of the events below. write each answer as a single fraction. p(a)= p(b)= p(a or b)= p(a and b)= p(a)+p(b)-p(a and b)= (b) select the probability that is equal to p(a)+p(b)-p(a and b). p(a and b) p(b) p(a or b) p(a)

Explanation:

Step1: Calculate total number of students

There are 11 students in total.

Step2: Calculate $P(A)$

Number of students in Music Club ($n(A)$) is 5. So $P(A)=\frac{n(A)}{n(\text{total})}=\frac{5}{11}$.

Step3: Calculate $P(B)$

Number of students in Tennis Club ($n(B)$) is 6. So $P(B)=\frac{n(B)}{n(\text{total})}=\frac{6}{11}$.

Step4: Calculate $P(A\text{ or }B)$

Number of students in either Music or Tennis Club ($n(A\cup B)$) is 8. So $P(A\text{ or }B)=\frac{n(A\cup B)}{n(\text{total})}=\frac{8}{11}$.

Step5: Calculate $P(A\text{ and }B)$

Number of students in both Music and Tennis Club ($n(A\cap B)$) is 3. So $P(A\text{ and }B)=\frac{n(A\cap B)}{n(\text{total})}=\frac{3}{11}$.

Step6: Calculate $P(A)+P(B)-P(A\text{ and }B)$

$P(A)+P(B)-P(A\text{ and }B)=\frac{5}{11}+\frac{6}{11}-\frac{3}{11}=\frac{5 + 6- 3}{11}=\frac{8}{11}$.

Answer:

$P(A)=\frac{5}{11}$
$P(B)=\frac{6}{11}$
$P(A\text{ or }B)=\frac{8}{11}$
$P(A\text{ and }B)=\frac{3}{11}$
$P(A)+P(B)-P(A\text{ and }B)=\frac{8}{11}$
(b) $P(A\text{ or }B)$