This Exam P sample reference tests Inclusion–Exclusion. Inclusion–exclusion gives probability 0.48 for at least one of the three events, so the complementary probability is 0.52, choice D.
Let G, B, and S denote the three events, and let U be the event that at least one of them occurs.
U=G∪B∪S
Model
Model
Apply the three-event inclusion–exclusion formula: add the marginal probabilities, subtract all pairwise overlaps, and restore the triple overlap once.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
BThe value 0.36 is the sum 0.14+0.10+0.12 of the three pairwise overlaps. That sum is only one component of inclusion–exclusion, not the requested probability.
EOmitting the +0.08 triple-overlap restoration gives a union probability of 0.40 and hence the incorrect complement 0.60.
Original practice · fully worked
Original variant: exactly one detector flags a junction
Three monitoring teams review the same set of road junctions. Historical proportions flagged by teams A, B, and C are 0.35, 0.25, and 0.20. Pairwise overlap proportions are 0.10, 0.08, and 0.05, and all three teams flag 0.02. Find the proportion flagged by exactly one team.
A 0.30
B 0.34
C 0.40
D 0.46
E 0.60
Variant answer in brief
Exactly-one membership equals the marginal sum minus twice the pairwise-overlap sum plus three times the triple overlap, giving 0.40, choice C.
Setup
Setup
Let p-one denote the proportion of junctions flagged by exactly one of the three teams.
p1=Pr(exactly one flag)
Model
Model
In the marginal sum, an exactly-two junction is counted twice and an exactly-three junction three times. Subtracting twice each pairwise overlap and restoring three times the triple overlap leaves exactly-one membership.
p1=∑Pr(Ai)−2∑Pr(Ai∩Aj)+3Pr(A∩B∩C)
Compute
Compute
The marginal probabilities sum to 0.80, the pairwise overlaps sum to 0.23, and the triple overlap is 0.02; substitution gives 0.40.
p1=0.80−2(0.23)+3(0.02)=0.40
Answer
Answer
Therefore 40% of junctions are flagged by exactly one team, which is 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.
