This Exam P sample reference tests Exponential Distribution. An exponential survival function is log-linear in the threshold. Comparing the two positive-payment probabilities gives a threshold ratio of 3/2, so the second deductible is 2.160, choice E.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AThe value 0.960 reverses the exponent ratio and computes 1.44(2/3). A smaller survival probability requires a larger, not smaller, deductible.
BThe value 1.152 multiplies 1.44 by the raw probability ratio 0.512/0.640. Exponential thresholds are proportional to log-survivals, not survival probabilities.
CThe value 1.728 treats the 20% relative decline from 0.640 to 0.512 as a 20% threshold increase. Survival probability does not change linearly with the deductible.
DThe value 1.800 multiplies 1.44 by 0.640/0.512=1.25. Using a raw inverse probability ratio still misses the logarithmic relationship.
Original practice · fully worked
Original variant: residual time for an exponential cooling alarm
The time from activation until a cooling alarm clears is exponentially distributed with mean 3 minutes. A monitor shows that the alarm has already remained active for 4 minutes. Find the conditional probability that it remains active for at least 5 additional minutes.
A 0.0498
B 0.1889
C 0.2636
D 0.8111
E 0.9502
Variant answer in brief
Exponential memorylessness resets the waiting clock after minute 4. The probability of another five minutes is exp(-5/3)=0.18888, choice B.
Setup
Setup
Let T be the total active time and convert the mean into a rate.
T∼Exponential(λ),λ=31
Model
Model
Remaining active for five more minutes means that the total exceeds nine, conditional on already exceeding four.
Pr(T>9∣T>4)=S(4)S(9)
Compute
Compute
Cancel the elapsed portion in the survival ratio.
e−4/3e−9/3=e−5/3=0.1888756028
Answer
Answer
The conditional probability is approximately 0.1889.
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