1. Stating the problem: There are 41 students in total. - 15 take Spanish (S) - 21 take French (F) - 7 take neither subject. 2. Find the number of students taking both Spanish and French. Total students taking at least one subject = Total students - Neither = 41 - 7 = 34 Using the formula for union of two sets: $$|S \cup F| = |S| + |F| - |S \cap F|$$ Substitute values: $$34 = 15 + 21 - |S \cap F|$$ Simplify: $$34 = 36 - |S \cap F|$$ So, $$|S \cap F| = 36 - 34 = 2$$ 3. Probability that a randomly chosen student studies French. Number of French students = 21 Total students = 41 Probability: $$P(F) = \frac{21}{41}$$ 4. Probability that a randomly chosen student does not study Spanish. Number of students studying Spanish = 15 Number of students not studying Spanish = 41 - 15 = 26 Probability: $$P(\text{not } S) = \frac{26}{41}$$ 5. Probability that a French student studies Spanish. Number of students studying both = 2 Number of French students = 21 Conditional probability: $$P(S|F) = \frac{|S \cap F|}{|F|} = \frac{2}{21}$$ 6. Probability that a Spanish student does not study French. Number of Spanish students = 15 Number of students studying both = 2 Number of Spanish students not studying French = 15 - 2 = 13 Conditional probability: $$P(\text{not } F|S) = \frac{13}{15}$$