This Exam P sample reference tests Exponential Memorylessness. Memorylessness converts the stated conditional survival into a 15-year survival probability of 0.40. Scaling the exponential survival to 50 years gives 0.047156, which rounds to choice B.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AAn exponential survival probability is strictly positive at every finite time, so rounding this result all the way to zero is not justified.
CThis is approximately 0.40 raised to the power 2.5, which treats 50 years as 2.5 blocks of 20 years instead of using the calibrated 15-year block.
DThis is 0.40², so it accounts for only two 15-year survival blocks, or 30 years, rather than 50 years.
EThis value exceeds 0.40² = 0.16, even though 50 years is longer than two calibrated 15-year blocks; it therefore violates monotonicity of survival.
Original practice · fully worked
Original variant: underground moisture sensor
The operating lifetime M of an underground moisture sensor is exponential. Given that a sensor has operated for at least 6 months, the probability that it operates for at least 18 months is 0.55. Calculate the probability that a new sensor operates for at least 42 months.
A 0.0503
B 0.0915
C 0.1234
D 0.1664
E 0.5500
Variant answer in brief
The conditional statement calibrates a 12-month survival probability of 0.55. Forty-two months is 3.5 such intervals, so the survival probability is 0.123387 and choice C.
Setup
Setup
Represent the exponential survival function by a monthly rate θ.
Pr(M≥t)=e−θt
Model
Model
Memorylessness removes the six months already survived and leaves a 12-month future interval.
Pr(M≥18∣M≥6)=Pr(M≥12)=e−12θ=0.55
Compute
Compute
Write the 42-month survival in units of the calibrated 12-month interval.
Pr(M≥42)=(e−12θ)42/12
0.553.5=0.1233870023…
Answer
Answer
A new sensor reaches 42 months with probability about 0.1234.
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