Original practice · fully worked
Original variant: mixture probability for transit time
Sixty percent of parcels travel express path A; every remaining parcel uses path B. Transit time is exponential with mean 2 days on A and 5 days on B. Find the probability delivery occurs between day 1 and day 4.
- A
0.3441 - B
0.4055 - E
0.4305 - D
0.5276 - C
0.5889
Answer and calculation
Weight each route's exponential interval probability: 0.6(exp(−0.5)-exp(−2))+0.4(exp(−0.2)-exp(−0.8))=0.4305.
Setup
Setup
Condition on the parcel's route. Route A has probability 0.60 and route B has probability 0.40.
P(A)=0.6,P(B)=0.4 Model
Model
For each route, compute the exponential probability of delivery between days one and four.
P(1<T<4)=0.6(e−1/2−e−2)+0.4(e−1/5−e−4/5) Compute
Compute
Weighting the two route-specific interval probabilities and adding them gives approximately 0.430478.
P(1<T<4)=0.6(0.471195)+0.4(0.369402)=0.430478 Answer
Answer
The delivery-interval probability is approximately 0.4305, selecting choice E.
0.4305(E) 