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
Answer and calculation
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.
0.40(A) 