1. **State the problem:** We have 65 Year 11 students studying at least one of Geography (G), History (H), and Religious Studies (R). Given the numbers in each region of the Venn diagram, we need to find the probability that a student who studies both History and Religious Studies does not study Geography. 2. **Identify the relevant sets and values:** - Students studying both History and Religious Studies (H \cap R) include those in: - H and R only (4 students) - G, H, and R (15 students) So, total students studying both History and Religious Studies: $$|H \cap R| = 4 + 15 = 19$$ 3. **Find students studying both History and Religious Studies but NOT Geography:** This is the number in the H and R only region: $$|H \cap R \cap G^c| = 4$$ 4. **Calculate the probability:** The probability that a student who studies both History and Religious Studies does not study Geography is: $$P = \frac{|H \cap R \cap G^c|}{|H \cap R|} = \frac{4}{19}$$ 5. **Final answer:** $$\boxed{\frac{4}{19}}$$ This means there is a 4 in 19 chance that a student studying both History and Religious Studies does not study Geography.