This Exam P sample reference tests Bayes’ Theorem. Expressing the three death rates relative to the light-smoker rate gives weighted contributions 0.25, 0.30, and 0.40. The heavy share is 0.42105, choice D.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AThe value 0.20 is the prior heavy-smoker share. The higher heavy-smoker death rate increases its share after conditioning on death.
BThe value 0.25 is the nonsmoker death weight 0.50(d/2), expressed relative to d; it is not a normalized posterior.
EThe value 0.57 is 0.40/(0.30+0.40) to rounding. It omits the nonsmoker death contribution 0.25d from the denominator.
Original practice · fully worked
Original variant: posterior risk from a sixfold signal
Ten percent of accounts belong to a high-risk class. A particular alert is six times as likely for a high-risk account as for any other account. Given that the alert appears, find the probability that the account is high risk.
A 0.40
B 0.10
C 0.25
D 0.50
E 0.60
Variant answer in brief
Alert weights are proportional to 6(0.10)=0.60 and 1(0.90)=0.90. The high-risk posterior is 0.60/1.50=0.40, choice A.
Setup
Setup
Let q be the alert probability for an ordinary account; the high-risk alert probability is then 6q.
Pr(A∣H)=6q,Pr(A∣Hc)=q
Model
Model
Multiply each likelihood by its class prior and normalize the high-risk alert weight.
Pr(H∣A)=6q(0.10)+q(0.90)6q(0.10)
Compute
Compute
The weights are proportional to 6(0.10)=0.60 and 1(0.90)=0.90, so the high-risk posterior is 0.60/1.50=0.40.
Pr(H∣A)=0.60/1.50=0.40
Answer
Answer
Given the alert, the account is high risk with probability 0.40, which is choice A.
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