This Exam P sample reference tests Set Probability. Partition the client population by mutually exclusive homeowners and renters coverage. Solving the resulting percentage equations gives 10% with both auto and renters, choice B.
Partition clients outside auto coverage into homeowners-only, renters-only, and neither. The supplied percentages determine each of these disjoint cells.
P(Ac)=0.36
P(none)=0.17
P(H∩Ac)=0.11
Model
Model
Subtract the neither and homeowners-only cells from the 36% without auto to obtain 8% renters-only. Let total homeowners coverage be twice total renters coverage.
P(R∩Ac)=0.36−0.17−0.11=0.08
H=2R
Compute
Compute
Express the two auto intersections by subtracting the known non-auto cells from their coverage totals. The 35% double-coverage condition gives total renters coverage 18%.
(H−0.11)+(R−0.08)=0.35
3R=0.54
R=0.18
P(A∩R)=0.18−0.08=0.10
Answer
Answer
Subtracting the 8% renters-only cell leaves 10% with both auto and renters coverage, choice B.
10%(B)
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These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AThe renters-only cell is 8%, which is near but not equal to choice A's 7%; no standard exact operation producing 7% was identified.
DThe value 25% is the homeowners-and-auto intersection: total homeowners coverage 36% minus 11% homeowners-only.
Original practice · fully worked
Original variant: overlap of streaming memberships
Customers may subscribe to Music, News, or Sports, but News and Sports are mutually exclusive. Twenty percent have none, 60% have Music, 15% have News without Music, and 30% have exactly two services. Total News membership is twice total Sports membership. What percentage have both Music and Sports?
A 5.00%
C 11.67%
B 15.00%
D 20.00%
E 30.00%
Variant answer in brief
Let total Sports be s and total News 2s. The double-membership total is (2s-0.15)+(s-r_only); using the outside-Music partition identifies the requested overlap.
Setup
Setup
Partition customers outside Music into none, News-only, and Sports-only, using the mutual exclusion of News and Sports.
P(Mc)=0.40
P(none)=0.20
P(N∩Mc)=0.15
Model
Model
The outside-Music partition gives 5% Sports-only. Let total News membership be twice total Sports membership.
P(S∩Mc)=0.40−0.20−0.15=0.05
N=2S
Compute
Compute
Express both two-service cells by subtracting the known single-service cells. The stated 30% total then gives Sports membership 1/6 and Music-plus-Sports membership 11.67%.
(2S−0.15)+(S−0.05)=0.30
3S=0.50
S=61
P(M∩S)=61−0.05=0.116667
Answer
Answer
Therefore 11.67% have both Music and Sports, corresponding to choice C.
The 3108-page Probability Proof Manual reorganizes 718 verified Exam P solutions by syllabus skill and adds formula proofs, error patterns, and original worked practice.
