This Exam P sample reference tests Bayes’ Theorem. Let the smoking rate without the condition be q; then the two Bayes weights are 0.50q and 0.75q. Their ratio gives posterior 0.40, choice C.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AThe value 1/4 is the prior probability of the condition and ignores the doubled smoking likelihood.
BThe value 1/3 is the prior odds 0.25/0.75 treated as a probability; it does not normalize the two smoking weights.
DThe value 1/2 is the unnormalized condition-and-smoking weight after q is canceled, not the posterior ratio.
EThe value 2/3 normalizes the likelihood ratio 2:1 without using the unequal prior shares 0.25 and 0.75.
Original practice · fully worked
Original variant: posterior membership from a likelihood ratio
Thirty percent of devices use controller C. A warning is three times as likely on devices using C as on other devices. Given that a warning appears, find the probability that the device uses controller C.
A 0.3000
B 0.5000
C 0.5625
D 0.7000
E 0.9000
Variant answer in brief
Warning weights are proportional to 3(0.30) and 1(0.70). The controller-C share is 0.90/1.60=0.5625, choice C.
Setup
Setup
Let q be the warning probability for devices not using controller C; devices using C have warning probability 3q.
Pr(W∣C)=3q,Pr(W∣Cc)=q
Model
Model
Multiply the two warning likelihoods by the controller priors and normalize the controller-C weight.
Pr(C∣W)=3q(0.30)+q(0.70)3q(0.30)
Compute
Compute
After q cancels, the controller-C and other-controller weights are 0.90 and 0.70, so the posterior is 0.90/1.60=0.5625.
Pr(C∣W)=0.90/1.60=0.5625
Answer
Answer
Given a warning, the probability that the device uses controller C is 0.5625, which is choice C.
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