1. **Problem statement:** We have 60 students: 35 study French, 45 study Spanish, and 27 study both. Find: a) Probability a student studies only one subject. b) Probability a student studies French given they study Spanish. c) Probability a student studies Spanish given they do not study French. 2. **Formulas and rules:** - Probability of an event $E$ is $P(E) = \frac{\text{number of favorable outcomes}}{\text{total outcomes}}$. - For two events $A$ and $B$, $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. - Conditional probability: $P(A|B) = \frac{P(A \cap B)}{P(B)}$. - Complement rule: $P(A') = 1 - P(A)$. 3. **Calculations:** **a) Probability a student studies only one subject:** - Number studying only French = $35 - 27 = 8$ - Number studying only Spanish = $45 - 27 = 18$ - Total studying only one subject = $8 + 18 = 26$ - Probability = $\frac{26}{60}$ **b) Probability a student studies French given they study Spanish:** - $P(\text{French} \cap \text{Spanish}) = \frac{27}{60}$ - $P(\text{Spanish}) = \frac{45}{60}$ - Conditional probability: $$ P(\text{French} | \text{Spanish}) = \frac{P(\text{French} \cap \text{Spanish})}{P(\text{Spanish})} = \frac{\frac{27}{60}}{\frac{45}{60}} = \frac{27}{45} = \frac{3}{5} $$ **c) Probability a student studies Spanish given they do not study French:** - Number not studying French = $60 - 35 = 25$ - Number studying Spanish but not French = $45 - 27 = 18$ - Probability: $$ P(\text{Spanish} | \text{French}') = \frac{18}{25} $$ 4. **Final answers:** - a) $\boxed{\frac{26}{60} = \frac{13}{30}}$ - b) $\boxed{\frac{3}{5} = 0.6}$ - c) $\boxed{\frac{18}{25} = 0.72}$