# Question A survey of a group's viewing habits over the last year revealed the following information: - (i) 28% watched gymnastics - (ii) 29% watched baseball - (iii) 19% watched soccer - (iv) 14% watched gymnastics and baseball - (v) 12% watched baseball and soccer - (vi) 10% watched gymnastics and soccer - (vii) 8% watched all three sports Calculate the percentage of the group that watched none of the three sports during the last year. ## Answer Choices (A) 24% (B) 36% (C) 41% (D) 52% (E) 60% ## Solution Let $G$, $B$, and $S$ represent the events that a person watched gymnastics, baseball, and soccer respectively. Given: - $P(G) = 0.28$ - $P(B) = 0.29$ - $P(S) = 0.19$ - $P(G \cap B) = 0.14$ - $P(B \cap S) = 0.12$ - $P(G \cap S) = 0.10$ - $P(G \cap B \cap S) = 0.08$ Using the Inclusion-Exclusion Principle: $P(G \cup B \cup S) = P(G) + P(B) + P(S) - P(G \cap B) - P(B \cap S) - P(G \cap S) + P(G \cap B \cap S)$ The probability of watching at least one of the three sports is: $P(G \cup B \cup S) = 0.28 + 0.29 + 0.19 - 0.14 - 0.12 - 0.10 + 0.08 = 0.48$ Therefore, the probability of watching none of the three sports is: $P(\text{none}) = 1 - P(G \cup B \cup S) = 1 - 0.48 = 0.52 = 52\%$ **Answer: (D)**