This Exam P sample reference tests Independence. Independence lets the two exponential failure-within probabilities multiply. The product expands to the expression in official choice C.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AChoice A introduces an extra scaling factor not present in the multiplication rule for independent events.
BChoice B retains only a joint survival term and omits the other terms from expanding the product of two cumulative probabilities.
DChoice D combines the two exponential survival terms with the wrong exponent.
EChoice E assigns weights to the two event probabilities instead of multiplying them under independence.
Original practice · fully worked
Original variant: two independent repair deadlines
A software repair time is exponential with mean 4 hours, and an independent hardware repair time is exponential with mean 6 hours. Find the probability that software finishes within 3 hours while hardware takes more than 5 hours.
A 0.1890
D 0.2292
C 0.2741
B 0.3102
E 0.5276
Variant answer in brief
Independence gives (1-exp(-3/4))exp(-5/6)=0.2292.
Setup
Setup
The software event is completion within 3 hours; the independent hardware event is a duration longer than 5 hours.
P(S≤3)=1−e−3/4
P(H>5)=e−5/6
Model
Model
Use the software cumulative probability and the hardware survival probability, then multiply them by independence.
P(S≤3,H>5)=(1−e−3/4)e−5/6
Compute
Compute
The component probabilities are 0.5276334 and 0.4345982, whose product is 0.229292.
P=0.5276334(0.4345982)=0.229292
Answer
Answer
The joint probability is 0.2292, corresponding to choice D.
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.
