This Exam P sample reference tests Inclusion–Exclusion. This is an inclusion–exclusion bounds problem. Writing the unknown overlap as q makes the probability of neither event 0.1+q; the feasible overlap runs from 0 to 0.3, so the requested range is 0.1 to 0.4 and choice C is correct.
Let the two indicated actions be events A and B, and denote their unknown overlap probability by q.
Pr(A)=0.30,Pr(B)=0.60,q=Pr(A∩B)
Model
Model
Use inclusion–exclusion for the union, then take its complement to express the probability p of neither action.
Pr(A∪B)=0.30+0.60−q
p=1−Pr(A∪B)=0.10+q
Compute
Compute
The Fréchet bounds restrict the intersection to the interval from zero through the smaller marginal probability. Substituting those endpoints gives the full feasible range for p.
max(0,0.30+0.60−1)≤q≤min(0.30,0.60)
0≤q≤0.30⟹0.10≤p≤0.40
Answer
Answer
Every intermediate overlap is feasible, so the probability of neither action can be any value in the closed interval shown.
0.10≤p≤0.40(C)
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These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AThis interval can arise from giving the overlap the wrong sign in the complement, writing 0.10-q, and then truncating the resulting negative values at zero.
BThe value 0.10 is only the endpoint obtained when the two events are disjoint; the information does not force their intersection to be zero.
DThe value 0.30 corresponds to selecting q=0.20 in p=0.10+q, but no condition singles out that particular overlap.
EThe range 0.70 to 1.00 belongs to the complement of the intersection, 1-q, which means 'not both' rather than 'neither.'
Original practice · fully worked
Original variant: overnight processing features
For an overnight data-processing batch, event A means that edge compression is used and event B means that redundant storage is used. Historical records give P(A)=0.55 and P(B)=0.35, but they do not specify the dependence between the events. Determine the full possible range of the probability that exactly one of the two features is used.
A 0.000 to 0.350
B 0.200 to 0.550
C 0.200 to 0.900
D 0.550 to 0.900
E 0.650
Variant answer in brief
If q is the probability that both features are used, the exactly-one probability is 0.90-2q. Since q can range from 0 to 0.35, the answer ranges from 0.20 to 0.90, selecting choice C.
Setup
Setup
Represent the unspecified dependence by the intersection probability q.
q=Pr(A∩B)
0≤q≤min(0.55,0.35)=0.35
Model
Model
Exactly one feature is used in either A without B or B without A. Each marginal contains the intersection once.
Pr(exactly one)=Pr(A)+Pr(B)−2Pr(A∩B)
Pr(exactly one)=0.90−2q
Compute
Compute
The expression decreases with q, so its endpoints occur at the two feasible endpoint overlaps.
q=0⟹Pr(exactly one)=0.90
q=0.35⟹Pr(exactly one)=0.20
Answer
Answer
The possible exactly-one probability fills the interval from 0.20 through 0.90.
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