1. Stating the problem: We have 255 students studying history (HIS), geography (GEO), and mathematics (MATH) with given numbers of students studying each and their intersections. We want to find probabilities based on these numbers. 2. Given values: - Total students, $N = 255$ - $|HIS| = 90$ - $|GEO| = 95$ - $|MATH| = 165$ - $|HIS \cap GEO| = 18$ - $|HIS \cap MATH| = 75$ - $|GEO \cap MATH| = 60$ - $|HIS \cap GEO \cap MATH| = 15$ 3. From the Venn diagram description in the user message, we note parts: - Only History = 70 - Intersect HIS & GEO = 30 - Right-only GEO = 170 (seems inconsistent with total GEO; we rely on numeric probabilities given) - HIS & MATH = 50 - All three = 15 - GEO & MATH = 60 - Only MATH = 85 However, we rely on problem's explicitly stated numbers for calculation. 4. (a) Probability a student studies only history: From problem: number studying only history = 26 (given as 26/255) Probability $P(only\ HIS) = \frac{26}{255}$ 5. (b) Probability a student studies geography and mathematics but not history: Calculate $|GEO \cap MATH| - |HIS \cap GEO \cap MATH| = 60 - 15 = 45$ Probability $P(GEO \cap MATH \cap \neg HIS) = \frac{45}{255}$ Final answers: (a) $\boxed{\frac{26}{255}}$ (b) $\boxed{\frac{45}{255}}$