1. **Problem statement:** 100 students were surveyed about enjoying running (R), cycling (C), or swimming (S). We are given counts for each region in the Venn diagram and asked to find the number of students who enjoy none of these activities (x). 2. **Find x:** The total number of students is 100. The sum of all students who enjoy at least one activity plus those who enjoy none must equal 100. 3. **Sum all given values:** $$21 + 7 + 25 + 5 + 4 + 2 + 13 + x = 100$$ 4. **Calculate the sum of known values:** $$21 + 7 + 25 + 5 + 4 + 2 + 13 = 77$$ 5. **Set up equation for x:** $$77 + x = 100$$ 6. **Solve for x:** $$x = 100 - 77 = 23$$ 7. **Answer:** The number of students who enjoy none of these activities is **23**. --- **(b) Show that $P(C) = \frac{7}{20}$:** 1. Probability $P(C)$ is the number of students who enjoy cycling divided by total students. 2. Students who enjoy cycling are those in C only, C and R, C and S, and all three: $$21 + 7 + 2 + 5 = 35$$ 3. Total students = 100, so $$P(C) = \frac{35}{100} = \frac{7}{20}$$ --- **(c) Find $P(R \cup S)$:** 1. Use formula: $$P(R \cup S) = P(R) + P(S) - P(R \cap S)$$ 2. Calculate $P(R)$: $$25 + 7 + 4 + 5 = 41$$ 3. Calculate $P(S)$: $$13 + 4 + 2 + 5 = 24$$ 4. Calculate $P(R \cap S)$: $$4 + 5 = 9$$ 5. Calculate $P(R \cup S)$: $$\frac{41}{100} + \frac{24}{100} - \frac{9}{100} = \frac{56}{100} = \frac{14}{25}$$ --- **(d) Verify $P(R \cap C) \neq P(R) \times P(C)$:** 1. Calculate $P(R \cap C)$: $$7 + 5 = 12$$ 2. So, $$P(R \cap C) = \frac{12}{100} = \frac{3}{25}$$ 3. Recall from above: $$P(R) = \frac{41}{100}, \quad P(C) = \frac{7}{20}$$ 4. Calculate $P(R) \times P(C)$: $$\frac{41}{100} \times \frac{7}{20} = \frac{287}{2000} = 0.1435$$ 5. Compare: $$P(R \cap C) = 0.12 \neq 0.1435 = P(R) \times P(C)$$ --- **(e) Probability both students enjoy swimming:** 1. Number who enjoy swimming: $$13 + 4 + 2 + 5 = 24$$ 2. Probability one student enjoys swimming: $$\frac{24}{100} = \frac{6}{25}$$ 3. Probability both enjoy swimming (without replacement): $$\frac{24}{100} \times \frac{23}{99} = \frac{552}{9900}$$ 4. Simplify fraction: $$\frac{552}{9900} = \frac{46}{825}$$ --- **Final answers:** - (a) $x = 23$ - (b) $P(C) = \frac{7}{20}$ - (c) $P(R \cup S) = \frac{14}{25}$ - (d) $P(R \cap C) \neq P(R) \times P(C)$ - (e) Probability both enjoy swimming = $\frac{46}{825}$