Worked solution · Exam P
This Exam P sample reference tests Conditional Probability. The survival function is proportional to (20-x)², so the conditional tail ratio is (4/12)²=1/9, choice B.
Independent solution
Setup
Integrate the density from x to the upper endpoint 20 to obtain the survival function.
Model
Because X>16 is contained in X>8, the conditional probability is the ratio of the two survival probabilities.
Compute
The common factor cancels, leaving the squared distance ratio (4/12)²=1/9.
Answer
Therefore Pr(X>16 given X>8)=1/9, corresponding to choice B.
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30 original univariate random variables problems with complete solutions and wrong-choice analysis. Written for Exam P, not copied from the SOA set. One-time $19 PDF.
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Original practice · fully worked
A quality score X has density f(x)=x/8 for 0<x<4. Given that X exceeds 1, determine the probability that X is less than 2.
The CDF is x²/16. The mass from one to two is 3/16 and the mass above one is 15/16, giving 1/5, choice D.
Setup
Integrate the density from zero to x to obtain the cumulative distribution F(x)=x²/16.
Model
Within the condition X>1, the target event is the interval from 1 to 2; divide that interval's mass by the mass above 1.
Compute
The numerator is 3/16 and the denominator is 15/16, so the conditional probability is 3/15=0.20.
Answer
Thus Pr(X<2 given X>1)=0.20, which is choice D.
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These references share syllabus concepts or sit nearby in the official sequence. All 387 Exam P univariate random variables practice problems.
Drill the method: Exam P Conditional Probability problems
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