A tunnel of length of 800 m is to be driven. Part of the total length is through granite and balance is through sandstone. Granite can be tunneled at 10 m per day and sandstone at 20 m per day. Two mutually exclusive site conditions can be (i) A probability of 0.4 for 600 m granite with 200 m sandstone; (ii) Probability of 0.6 that each is 400 m long. The probabilistic estimate of total tunneling duration is
- A74 days
- B68 days
- C64 daysCorrect
- D60 days
Solution
Concept
Expected value of duration
Formula
E(T) = Σ p × T
Step-by-step solution
1. Condition (i), p = 0.4: 600 m granite = 60 days + 200 m sandstone = 10 days = 70 days. 2. Condition (ii), p = 0.6: 400 m granite = 40 days + 400 m sandstone = 20 days = 60 days. 3. Expected duration = 0.4 × 70 + 0.6 × 60 = 28 + 36 = 64 days.
Explanation
When site conditions are mutually exclusive, the expected duration is the probability-weighted average of the durations under each condition.
Shortcut
Expected value = p1 T1 + p2 T2.
Common mistake
Using 600/20 and 200/10 (swapping the rates).
Correct answer: C) 64 days