The Survey Logic Trap
A student council surveys 120 students about two optional activities: a movie night (M) and a games night (G). Results: 70 students attended M, 50 attended G, and 30 attended both. A council member concludes that 70 + 50 + 30 = 150, so 'more students attended than were surveyed, which is impossible.' Another member says the 30 who attended both were counted twice, so the total who attended at least one is 70 + 50 − 30 = 90. A third member claims that because 90 attended at least one, the remaining 30 attended neither. The council must decide which reasoning is valid and whether the survey is internally consistent.
1. Evaluate the first member's conclusion that 150 > 120 is 'impossible.' What logical error is being made?
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The first member added the counts of M, G, and both as if they were disjoint groups. But the 30 students who attended both are already included in the 70 who attended M and in the 50 who attended G. Adding them again double-counts them.
The sum 70 + 50 + 30 = 150 is not a count of distinct students; it is a meaningless total that counts the 'both' group three times (once in M, once in G, and once explicitly).
The error is a category mistake: treating overlapping sets as if they were mutually exclusive. The fact that 150 > 120 does not show the survey is impossible; it shows the arithmetic combined overlapping groups incorrectly. The correct count of distinct attendees is 70 + 50 − 30 = 90.
2. Verify the second member's calculation using the inclusion-exclusion principle, and determine how many students attended neither activity.
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Inclusion-exclusion for two sets: |M ∪ G| = |M| + |G| − |M ∩ G|.
|M ∪ G| = 70 + 50 − 30 = 90.
So 90 students attended at least one activity.
Total surveyed = 120, so students who attended neither = 120 − 90 = 30.
The second member's calculation is correct. The third member's conclusion (30 attended neither) follows correctly from the total. The survey is internally consistent: 90 + 30 = 120.
Note: the '30' who attended both and the '30' who attended neither are different groups that happen to share the same number; this is a coincidence, not an error.
3. The third member says 'the remaining 30 attended neither.' Is this a valid deduction, or does it assume something not stated?
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It is valid only if every surveyed student attended at least one of M or G, or neither - that is, the two categories 'attended at least one' and 'attended neither' are exhaustive and mutually exclusive. That is true by definition: every student either attended at least one activity or attended neither.
So the deduction 120 − 90 = 30 is valid, provided the survey counts are accurate and every student is in exactly one of those two categories. The assumption is that the survey responses are complete and consistent, which the problem states.
The reasoning is valid, but it depends on the premise that the 120 students are the full surveyed population and that no student is uncounted. If some students did not respond, the deduction would fail. So the deduction is conditionally valid, not unconditionally valid.