Original practice · fully worked
Original variant: posterior source after a mixed two-stage outcome
Thirty of every hundred processes run in mode A; all others use mode B. An individual trial succeeds with probability 0.80 in mode A and 0.20 in mode B. Two conditionally independent trials produce one success followed by one failure. Find the posterior probability of mode A.
- A
0.160 - B
0.200 - C
0.300 - D
0.500 - E
0.800
Answer and calculation
Both modes assign likelihood 0.16 to the ordered mixed outcome, so the evidence does not change the prior. The posterior remains 0.30, choice C.
Setup
Setup
Compute the ordered success-then-failure likelihood under each process mode.
LA=0.80(0.20)=0.16 LB=0.20(0.80)=0.16 Model
Model
Apply Bayes' formula by multiplying each equal likelihood by its mode prior and normalizing the mode-A weight.
Pr(A∣SF)=0.30LA+0.70LB0.30LA Compute
Compute
Both likelihoods equal 0.16, so they cancel from the Bayes ratio; the posterior remains the prior 0.30.
Pr(A∣SF)=0.048/(0.048+0.112)=0.30 Answer
Answer
The posterior probability of mode A is 0.300, which is choice C.
0.30(C) 