Question

question 8 of 10 this venn diagram shows the pizza topping preferences for 9 students. venn diagram: green circle (likes pepperoni) with joe, kalie, noah; overlap with red circle (likes olives) has maria, zack; red circle has julia; outside: bo, petal, owen let event a = the student likes pepperoni. let event b = the student likes olives. what is p(a or b)? a. \\(\frac{2}{3}\\) b. \\(\frac{2}{9}\\) c. \\(\frac{1}{6}\\) d. \\(\frac{1}{3}\\)

Explanation:

Answer:

To find \( P(A \text{ or } B) \), we use the principle of inclusion - exclusion for probability: \( P(A \text{ or } B)=P(A)+P(B)-P(A \text{ and } B) \).

First, we determine the number of students in each event:

• Total number of students \( n = 9 \).
• Number of students who like pepperoni (event \( A \)): The green circle (likes pepperoni) has Joe, Kalie, Noah, Maria, Zack. So \( n(A)=5 \).
• Number of students who like olives (event \( B \)): The red circle (likes olives) has Maria, Zack, Julia. So \( n(B) = 3 \).
• Number of students who like both pepperoni and olives (event \( A \) and \( B \)): The intersection of the two circles has Maria, Zack. So \( n(A \text{ and } B)=2 \).

Now we calculate the probabilities:

• \( P(A)=\frac{n(A)}{n}=\frac{5}{9} \)
• \( P(B)=\frac{n(B)}{n}=\frac{3}{9} \)
• \( P(A \text{ and } B)=\frac{n(A \text{ and } B)}{n}=\frac{2}{9} \)

Then, \( P(A \text{ or } B)=\frac{5}{9}+\frac{3}{9}-\frac{2}{9}=\frac{5 + 3-2}{9}=\frac{6}{9}=\frac{2}{3} \)

So the answer is A. \( \frac{2}{3} \)