Question

the following venn diagram shows the results of a survey of students in a math class. based on the venn diagram, whats the probability that a student did not eat a burger for lunch? provide an answer as a fraction.

Explanation:

Step1: Find total students

Total students = Burger only + Pizza only + Both + Neither = \(12 + 8 + 5 + 26\) = \(51\)? Wait, no, wait. Wait, the Venn diagram: Burger circle has 12 (only Burger), 5 (both), Pizza circle has 8 (only Pizza), and the rectangle has 26? Wait, no, maybe the 26 is the number of students who ate neither? Wait, the Venn diagram: the two circles (Burger and Pizza) inside a rectangle. So the regions: only Burger (12), only Pizza (8), both (5), and outside both (26). So total students = \(12 + 8 + 5 + 26\) = \(51\)? Wait, no, wait, maybe I misread. Wait, the problem: "the probability that a student did NOT eat a burger for lunch". So students who did NOT eat burger: only Pizza + neither. Only Pizza is 8, neither is 26. So number of students who did NOT eat burger: \(8 + 26 = 34\). Total students: \(12 + 5 + 8 + 26 = 51\)? Wait, no, wait, maybe the 26 is the number of students outside both? Let's check again.

Wait, the Venn diagram: Burger circle: 12 (only Burger), 5 (both). Pizza circle: 8 (only Pizza). The rectangle (total) has 26? No, maybe the numbers: 12 (Burger only), 5 (both), 8 (Pizza only), and 26 (neither). So total students: \(12 + 5 + 8 + 26 = 51\). Students who did NOT eat burger: those who ate Pizza only + neither. So \(8 + 26 = 34\). So probability is \( \frac{34}{51} \), simplify: divide numerator and denominator by 17: \( \frac{2}{3} \)? Wait, no, 34 ÷ 17 = 2, 51 ÷ 17 = 3. Wait, but wait, maybe I made a mistake in total. Wait, maybe the 26 is not neither? Wait, the problem says "the following Venn diagram shows the results of a survey of students in a math class". So the two circles: Burger and Pizza. So the regions:

• Only Burger: 12

• Both Burger and Pizza: 5

• Only Pizza: 8

• Neither Burger nor Pizza: 26

So total number of students: \(12 + 5 + 8 + 26 = 51\)

Number of students who did NOT eat a burger: students who ate Pizza only + students who ate neither. So \(8 + 26 = 34\)

Therefore, probability = \( \frac{\text{Number of students who did NOT eat burger}}{\text{Total number of students}} = \frac{34}{51} = \frac{2}{3} \) (after simplifying by dividing numerator and denominator by 17).

Wait, but let's check again. Wait, maybe the 26 is not neither? Wait, maybe the rectangle's number is the total? No, the Venn diagram: the two circles (Burger and Pizza) inside a rectangle. The numbers inside the circles: 12 (Burger only), 5 (both), 8 (Pizza only). The number outside both is 26. So total students: 12 + 5 + 8 + 26 = 51. Students who did NOT eat burger: those who are not in the Burger circle. So Pizza only (8) + neither (26) = 34. So probability is 34/51 = 2/3.

Step2: Simplify the fraction

\( \frac{34}{51} = \frac{34 \div 17}{51 \div 17} = \frac{2}{3} \)

Answer:

\(\boxed{\dfrac{2}{3}}\)