This Exam P sample reference tests Exponential Distribution. Exponential survival scales by time: survival through 80 days is 0.7⁽⁸⁰⁄⁵⁰⁾. Its complement is 0.43486, choice C.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
DThe value 0.57 is 0.70 raised to the 1.6 power, the 80-day survival probability rather than the requested failure probability.
Original practice · fully worked
Original variant: conditional survival after an elapsed operating time
The operating time T of a sensor follows an exponential distribution. Calibration shows that P(T exceeds 4 hours)=0.80. Given that the sensor is still operating at hour 4, determine the probability that it remains operating beyond hour 10.
A 0.64000
B 0.70000
C 0.80000
D 0.84000
E 0.71554
Variant answer in brief
Memorylessness turns the condition into six additional hours of survival. Scaling the four-hour survival gives 0.80⁽⁶⁄⁴⁾=0.71554, choice E.
Setup
Setup
Record that the exponential survival probability through four hours is 0.80.
e−4λ=0.80
Model
Model
By memorylessness, conditional survival from hour 4 to hour 10 is the unconditional probability of surviving six additional hours.
Pr(T>10∣T>4)=Pr(T>6)=e−6λ
Compute
Compute
Scale the four-hour survival to six hours by raising 0.80 to the 6/4 power, giving 0.715542.
e−6λ=(e−4λ)6/4=0.801.5=0.715542
Answer
Answer
The conditional probability of operating beyond hour 10 is 0.71554, which is choice E.
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