Original practice · fully worked
Original variant: infer seed-lot type from split assay results
A sealed seed lot is vigorous with probability 0.25 and ordinary with probability 0.75. Conditional on its type, two germination assays are independent. Each assay is positive with probability 0.60 for a vigorous lot and 0.20 for an ordinary lot. Exactly one of the two assays is positive. Calculate the probability that the lot is vigorous.
- A
0.1200 - B
0.2500 - C
0.3200 - D
0.3333 - E
0.4800
Answer and calculation
The exactly-one-positive likelihood is 0.48 for a vigorous lot and 0.32 for an ordinary lot. Prior weighting gives joint masses 0.12 and 0.24, so the vigorous posterior is 0.12/(0.12+0.24)=1/3 and choice D.
Setup
Setup
Let E denote exactly one positive result. Compute its two class-conditional likelihoods from the two possible positive positions.
Pr(E∣V)=2(0.60)(0.40)=0.48 Pr(E∣O)=2(0.20)(0.80)=0.32 Model
Model
Multiply each likelihood by its lot-type prior to obtain the two joint weights for the observed assay pattern.
Pr(V∩E)=0.25(0.48)=0.12 Pr(O∩E)=0.75(0.32)=0.24 Compute
Compute
Normalize the vigorous-lot joint weight over both possible lot types.
Pr(V∣E)=0.12+0.240.12=31 Answer
Answer
The posterior probability that the lot is vigorous is approximately 0.3333.
0.3333(D) 