Original practice · fully worked
Original variant: randomized scoring protocols
A calibration run uses protocol I with probability 1/2, protocol II with probability 1/3, and protocol III with probability 1/6. Under I, independent scores are drawn uniformly from {0,1,2} and {0,2}, then added. Under II, a score drawn uniformly from {1,3} is increased by 2. Under III, a score drawn uniformly from {0,1,2} is increased by 1. Calculate the probability that the final score equals 3.
- A
0.1389 - B
0.1667 - C
0.3056 - D
0.3333 - E
0.6944
Answer and calculation
The conditional target probabilities under protocols I, II, and III are 1/6, 1/2, and 1/3. Weighting by the protocol probabilities gives 11/36=0.305556, so choice C is correct.
Setup
Setup
Determine the target probability separately under each protocol.
qI=31⋅21=61 qII=21 qIII=31 Model
Model
The protocol label partitions the experiment, so its stated probabilities are the mixture weights.
Pr(S=3)=21qI+31qII+61qIII Compute
Compute
Substitute the three conditional probabilities.
Pr(S=3)=21⋅61+31⋅21+61⋅31 Pr(S=3)=3611=0.3055555556 Answer
Answer
The mixture probability rounds to 0.3056.
0.3056(C) 