This Exam P sample reference tests Set Operations. Resolving the inclusive intersections gives 23 in life-home only, 64 in auto-home only, and 88 in home only. The only-one total then gives 53 in life only and 129 in auto only; the remaining auto region has 18. Summing all seven regions gives 407, choice B.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AThe value 389 is the disjoint total with the 18-person life-auto-only region omitted.
CThe value 423 exceeds the sum of the seven determined regions by 16. It necessarily double-counts part of an overlap because every Venn region is already fixed.
DThe value 448 adds 41 people beyond the complete disjoint partition. It is incompatible with the stated auto and homeowner marginal totals.
EThe value 483 is 407+76. It treats the 76-person difference between two only-regions as an additional group, even though that difference is already built into their counts.
Original practice · fully worked
Original variant: recover a two-tag overlap
An archive contains 300 records. An urgent tag appears on 180 records, an external tag appears on 150 records, and exactly 170 records carry one of these tags but not both. Calculate the number of records carrying both tags.
A 30 records
B 80 records
C 150 records
D 170 records
E 250 records
Variant answer in brief
If x records carry both tags, then the urgent-only and external-only counts are 180-x and 150-x. Their sum is 170, giving x=80 and choice B.
Setup
Setup
Let x denote the intersection of the urgent and external tag sets.
∣U∩E∣=x
Model
Model
Express the exactly-one count as the sum of the two disjoint one-tag regions.
∣U∖E∣+∣E∖U∣=(180−x)+(150−x)=170
Compute
Compute
Solve the resulting linear equation for the overlap.
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