This Exam P sample reference tests Bayes’ Theorem. The ultra-preferred death weight is 0.0001 and the total death probability is 0.0071, producing 0.01408, choice D.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AThe value 0.0001 is the joint ultra-preferred death weight. It is the numerator of Bayes' formula before normalization.
BThe value 0.0010 is the conditional death rate within the ultra-preferred class, not the posterior class probability given death.
CThe value 0.0071 is the total death probability, which is the denominator of the posterior ratio.
EThe value 0.2817 is 0.002/0.0071 to rounding, the posterior share of the class with joint weight 0.40(0.005), not the ultra-preferred class.
Original practice · fully worked
Original variant: source inference from two repeated signals
A monitoring source is type A with probability 0.25 and type B otherwise. On each independent trial it emits a red signal with probability 0.60 if type A and 0.20 if type B. Two consecutive red signals are observed. Find the posterior probability of type A.
A 0.250
B 0.750
C 0.600
D 0.500
E 0.900
Variant answer in brief
The two-red weights are 0.25(0.60²)=0.09 and 0.75(0.20²)=0.03. Type A therefore has posterior probability 0.75, choice B.
Setup
Setup
For each source type, multiply its prior probability by the conditional likelihood of two consecutive red signals.
wA=0.25(0.60)2=0.09
wB=0.75(0.20)2=0.03
Model
Model
Normalize the type-A two-red weight by the sum of the type-A and type-B weights.
Pr(A∣RR)=wA+wBwA
Compute
Compute
The weights are 0.09 and 0.03, so the posterior probability of type A is 0.09/(0.09+0.03)=0.75.
Pr(A∣RR)=0.09/0.12=0.75
Answer
Answer
Given two red signals, the source is type A with probability 0.75, which is choice B.
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