This Exam P sample reference tests Exponential Distribution. The equipment remains operational only when both independent exponential components exceed the five-year threshold. Multiplying their survival probabilities gives 0.054114, which rounds to choice B.
How to solve this Exponential Distribution question
Setup
Setup
Let P and S be the two component lifetimes. Convert their respective means of 3 and 4 years to exponential rates.
P∼Exp(1/3),S∼Exp(1/4)
Pr(P>t)=e−t/3,Pr(S>t)=e−t/4
Model
Model
A series system survives the target duration precisely when both components survive it. Independence turns the joint survival probability into a product.
Pr(min(P,S)≥5)=Pr(P≥5)Pr(S≥5)
Compute
Compute
Insert the two exponential survival probabilities and combine their exponents.
Pr(min(P,S)≥5)=e−5/3e−5/4
e−5/3−5/4=e−35/12=0.0541137662…
Answer
Answer
The system survival probability is about 0.05 at the precision of the listed answers.
Pr(min(P,S)≥5)≈0.05(B)
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These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AA continuous lifetime has probability zero of equaling one exact time, but the requested event is survival through a threshold, not equality at that instant.
CThe value exp(−5/3)=0.188876 uses only the component with mean 3 and omits the second required component.
DThe value exp(−5/4)=0.286505 uses only the component with mean 4 and omits the first required component.
EThe value (1-exp(−5/3))(1-exp(−5/4))=0.578733 is the probability that both components have failed by year 5, the opposite joint event.
Original practice · fully worked
Original variant: remote greenhouse controls
A remote greenhouse stays within its safe temperature range only while both a circulation fan and a thermostat remain operational. Their independent operating times are exponential with means 6 months and 10 months. Calculate the probability that both devices last through an 8-month unattended season.
A 0.118
B 0.264
C 0.406
D 0.449
E 0.594
Variant answer in brief
Both independent devices must survive eight months, so their exponential survival probabilities multiply. The result is 0.118442, which selects choice A.
Setup
Setup
Represent the fan and thermostat lifetimes by F and T and write their survival functions from the stated means.
Pr(F>t)=e−t/6,Pr(T>t)=e−t/10
Model
Model
Safe operation for the entire season requires the intersection of the two eight-month survival events.
Pr(F>8,T>8)=Pr(F>8)Pr(T>8)
Compute
Compute
Multiply the independent survival probabilities and combine the rates.
Pr(F>8,T>8)=e−8/6e−8/10
e−32/15=0.1184418290…
Answer
Answer
The greenhouse remains protected for the full season with probability approximately 0.118.
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