1. **State the problem:** We have 100 pupils surveyed at a sports centre. They can use either the swimming pool (S), the gym (G), or neither. Given: - Total pupils, $|\xi| = 100$ - Pupils using swimming pool, $|S| = 59$ - Pupils using gym, $|G| = 60$ - Pupils using neither, $|S^c \cap G^c| = 8$ We need to complete the Venn diagram and find the probability $P(S \cap G')$, i.e., pupils who use the swimming pool but not the gym. 2. **Use the formula for union of two sets:** $$|S \cup G| = |S| + |G| - |S \cap G|$$ 3. **Calculate $|S \cup G|$:** Since 8 pupils use neither, the number using at least one is: $$|S \cup G| = 100 - 8 = 92$$ 4. **Find the intersection $|S \cap G|$:** $$92 = 59 + 60 - |S \cap G|$$ $$|S \cap G| = 59 + 60 - 92 = 119 - 92 = 27$$ 5. **Find pupils using only swimming pool $|S \cap G'|$:** $$|S \cap G'| = |S| - |S \cap G| = 59 - 27 = 32$$ 6. **Find pupils using only gym $|G \cap S'|$:** $$|G \cap S'| = |G| - |S \cap G| = 60 - 27 = 33$$ 7. **Complete the Venn diagram:** - Only swimming pool: 32 - Only gym: 33 - Both: 27 - Neither: 8 8. **Calculate the probability $P(S \cap G')$:** $$P(S \cap G') = \frac{|S \cap G'|}{|\xi|} = \frac{32}{100} = 0.32$$ **Final answer:** $$\boxed{0.32}$$