This Exam P sample reference tests Conditional Probability. This problem combines conditional probability with a three-event union. Regrouping the union makes its probability 0.80-0.80(0.15r)+0.17r; setting this equal to one gives r=4, which is choice D.
How to solve this Conditional Probability question
Setup
Setup
Group the full union as event B together with the union of A and C. The overlap between those two grouped events simplifies because B and C cannot occur together.
B∩(A∪C)=(A∩B)∪(B∩C)=A∩B
Pr(A∩B)=Pr(B)Pr(A∣B)
Model
Model
Apply the two-set addition rule to the grouped events and substitute the supplied expressions.
Pr(A∪B∪C)=Pr(B)+Pr(A∪C)−Pr(A∩B)
1=0.80+0.17r−0.80(0.15r)
Compute
Compute
Combine the two coefficients of r and isolate the parameter.
1=0.80+(0.17−0.12)r=0.80+0.05r
r=0.050.20=4
Answer
Answer
The parameter value satisfying all of the event relationships is four.
r=4(D)
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These notes identify the calculation error associated with each wrong letter when that error is reproducible.
ASetting r to zero leaves the grouped union probability at 0.80, so it cannot satisfy the stated total probability of one.
BThe value 20/17 comes from solving 0.80+0.17r=1 and therefore treating A union C as disjoint from B; it omits the A intersection B overlap.
CThe value 100/29 comes from the incorrect equation 1=0.12r+0.17r, which drops the standalone 0.80 mass and adds the overlap rather than subtracting it.
ETreating 0.15r as P(A intersection B), instead of as the conditional probability P(A given B), produces r=10 and falsely suggests that none of A through D applies.
Original practice · fully worked
Original variant: diagnostic signals
Every diagnostic batch produces at least one of three signals: drift D, latency L, or thermal T. Records show that P(D union T)=0.58 and P(D given L)=0.25. A latency signal and a thermal signal can never occur in the same batch. Calculate P(L).
A 0.315
B 0.420
C 0.560
D 0.580
E 0.670
Variant answer in brief
Because latency and thermal signals are mutually exclusive, the overlap of L with D union T is just D intersection L. The full-union equation becomes 1=b+0.58-0.25b, giving b=0.56 and choice C.
Setup
Setup
Write b for the latency probability. The conditional drift rate among latency batches determines their intersection.
b=Pr(L)
Pr(D∩L)=Pr(L)Pr(D∣L)=0.25b
Model
Model
Combine L with D union T. Since L and T are disjoint, their grouped overlap contains only D intersection L.
Pr(L∩(D∪T))=Pr(D∩L)=0.25b
1=Pr(L)+Pr(D∪T)−Pr(L∩(D∪T))
Compute
Compute
Insert the known union probability and solve the one-variable equation.
1=b+0.58−0.25b=0.58+0.75b
b=0.750.42=0.56
Answer
Answer
The latency signal occurs in 56% of diagnostic batches.
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.
