This Exam P sample reference tests Law of Total Probability. This is a two-branch mixture whose unknown branch probability is determined by a cumulative probability. Solving the mixture equation gives a weighted-die branch probability of 0.666, after which the probability of rolling a four is 0.203667 and choice C.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AThe value 0.167 is the fair-die probability of a four. It ignores the branch using the weighted die.
BThe value 0.194 is the simple average of 1/6 and 2/9, which incorrectly assumes that the two branches are equally likely.
DThe value 0.537 is 1-0.463, the probability of a value above three; it includes faces five and six as well as face four.
EThe value 0.667 is the inferred probability of selecting the weighted branch, not the unconditional probability of the requested face.
Original practice · fully worked
Original variant: adaptive severity labels
A diagnostic platform routes a record to an adaptive classifier with probability r and to a legacy classifier otherwise. The adaptive classifier assigns severity labels 1 through 5 with probabilities 0.10, 0.10, 0.20, 0.20, and 0.40. The legacy classifier assigns the five labels uniformly. Historical output has P(S at most 3)=0.47. Calculate P(S=5).
A 0.200
B 0.260
C 0.300
D 0.330
E 0.650
Variant answer in brief
The adaptive and legacy cumulative probabilities through label three are 0.40 and 0.60. Their mixture equals 0.47 only when r=0.65, and the resulting probability of label five is 0.33, selecting choice D.
Setup
Setup
Compute the conditional cumulative probabilities through label three for the two classifiers.
Pr(S≤3∣adaptive)=0.10+0.10+0.20=0.40
Pr(S≤3∣legacy)=3(0.20)=0.60
Model
Model
Use the observed cumulative probability to identify the routing proportion.
0.47=0.40r+0.60(1−r)
Compute
Compute
Solve the mixture equation and then combine the two label-five probabilities.
r=0.200.60−0.47=0.65
Pr(S=5)=0.65(0.40)+0.35(0.20)=0.33
Answer
Answer
The platform assigns severity label five with probability 0.330.
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