Question

using a venn diagram to understand the addition rule for probability
e venn diagram below shows the 16 students in ms. changs class.
ie diagram shows the memberships for the chess club and the soccer club.
note that \michael\ is outside the circles since he is not a member of either club.
one student from the class is randomly selected.
let a denote the event \the student is in the chess club.\
let b denote the event \the student is in the soccer club.\
(a) find the probabilities of the events below.
write each answer as a single fraction.
$p(a)=\square$
$p(b)=\square$
$p(a\text{ or }b)=\square$
$p(a\text{ and }b)=\square$
$p(a)+p(b)-p(a\text{ and }b)=\square$

Explanation:

Step1: Count total students

Total students = 16

Step2: Calculate P(A)

Count students in Chess Club (A): Deshaun, Shen, Frank, Heather, Jane, Ann, Lashonda, Chang, Joe, Melissa = 10 students.
$P(A) = \frac{10}{16} = \frac{5}{8}$

Step3: Calculate P(B)

Count students in Soccer Club (B): Ryan, Charmaine, Lisa, Reuben, Chau, Lashonda, Chang, Joe, Melissa = 9 students.
$P(B) = \frac{9}{16}$

Step4: Calculate P(A and B)

Count students in both clubs: Lashonda, Chang, Joe, Melissa = 4 students.
$P(A \text{ and } B) = \frac{4}{16} = \frac{1}{4}$

Step5: Calculate P(A or B)

Use addition rule: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$
$P(A \text{ or } B) = \frac{10}{16} + \frac{9}{16} - \frac{4}{16} = \frac{15}{16}$

Step6: Verify addition rule

Substitute values into the formula:
$P(A) + P(B) - P(A \text{ and } B) = \frac{10}{16} + \frac{9}{16} - \frac{4}{16} = \frac{15}{16}$

Answer:

$P(A) = \frac{5}{8}$
$P(B) = \frac{9}{16}$
$P(A \text{ or } B) = \frac{15}{16}$
$P(A \text{ and } B) = \frac{1}{4}$
$P(A) + P(B) - P(A \text{ and } B) = \frac{15}{16}$