This Exam P sample reference tests Competing Geometric Waiting Times. At the first round with any claim, the favorable outcome has probability 0.08 and all terminating outcomes total 0.28, so the race probability is 2/7=0.2857, choice A.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
CThe value one third treats the three terminating outcomes—Rahul only, the other examiner only, and both—as equally likely, although their probabilities differ.
DThe value 0.3571 uses Rahul's marginal claim probability as the numerator and therefore counts simultaneous claims as Rahul finishing first.
EThe value 0.4000 discards simultaneous-claim rounds instead of treating them as terminating ties.
Original practice · fully worked
Original variant: which inspector finds a defect first
Two inspectors test independent streams of parts. Each part tested by inspector A is defective with probability 0.15, while each tested by inspector B is defective with probability 0.25. They test simultaneously until their first defects. Find the probability A stops first.
A 0.2500
B 0.3103
C 0.3625
D 0.4138
E 0.6375
Variant answer in brief
A-only success has probability 0.15(0.75)=0.1125, while a round terminates with probability 1-0.85(0.75)=0.3625. Their ratio is 0.3103, choice B.
Setup
Setup
In a simultaneous test round, neither inspector finds a defect with probability 0.6375. After such a round, the same race restarts.
q=(0.85)(0.75)=0.6375
Model
Model
Inspector A stops first only when A finds a defect and B does not in the first round containing at least one defect.
pAonly=(0.15)(0.75)=0.1125
Compute
Compute
The first terminating round has probability 0.3625, while the favorable A-only outcome has probability 0.1125. Their ratio is 0.310345.
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