Geometry

Question Analysis
This question requires analyzing a Venn Diagram to calculate the probability of a compound event where a student has exactly two hobbies.
The main focus is on correctly identifying and summing up the relevant regions in the Venn Diagram that represent students with pairs of hobbies, and then calculating the probability.

Key Concept Explanation
The probability of an event is determined by the ratio of the number of favorable outcomes to the total number of outcomes.
For the event of having exactly two hobbies, the favorable outcomes are the students in the regions “R and P only”, “R and D only”, and “P and D only”.
The total number of outcomes is the sum of all the numbers in the Venn Diagram, representing the total number of students with these hobbies.

Step - by - Step Solution
Identify the relevant regions: The regions representing students with exactly two hobbies are “R and P only” (8 students), “R and D only” (7 students), and “P and D only” (6 students).
Calculate the number of favorable outcomes: Sum up the number of students in these regions: .
Calculate the total number of students: Add up all the numbers in the Venn Diagram:

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