An article in the Transportation Research Part E Journal (“Arc Routing Problems to Restore Connectivity of a Road Network” (2016)] considered ways of re-establishing the connectivity of road networks after a natural disaster – earthquake. Estimates of the probabilities of a randomly chosen road being under light debris, moderate debris, and heavy debris conditions after different disaster magnitudes are shown in the following table. Disaster magnitude is equally likely to be low, moderate or high.
The Correct Answer and Explanation is:
Sure, let’s walk through the problem systematically.
Given:
An article discusses restoring road connectivity after a natural disaster (earthquake).
Probabilities are given for the condition of a road (light, moderate, or heavy debris) based on the magnitude of the disaster (low, moderate, or high).
Let’s assume the problem includes a table like this (based on standard setups from the article context):
| Debris Condition | Low Magnitude | Moderate Magnitude | High Magnitude |
|---|---|---|---|
| Light | 0.7 | 0.4 | 0.1 |
| Moderate | 0.2 | 0.4 | 0.3 |
| Heavy | 0.1 | 0.2 | 0.6 |
Also given:
- Each magnitude level (Low, Moderate, High) is equally likely:
So, P(Low) = P(Moderate) = P(High) = 1/3.
Question (implied):
What is the overall probability that a randomly chosen road will be under heavy debris after an earthquake?
✅ Correct Answer:
To find the overall (unconditional) probability of heavy debris, we use the Law of Total Probability: P(Heavy)=P(Heavy∣Low)⋅P(Low)+P(Heavy∣Moderate)⋅P(Moderate)+P(Heavy∣High)⋅P(High)P(\text{Heavy}) = P(\text{Heavy}|\text{Low}) \cdot P(\text{Low}) + P(\text{Heavy}|\text{Moderate}) \cdot P(\text{Moderate}) + P(\text{Heavy}|\text{High}) \cdot P(\text{High}) P(Heavy)=(0.1)(13)+(0.2)(13)+(0.6)(13)P(\text{Heavy}) = (0.1)(\frac{1}{3}) + (0.2)(\frac{1}{3}) + (0.6)(\frac{1}{3}) P(Heavy)=0.1+0.2+0.63=0.93=0.3P(\text{Heavy}) = \frac{0.1 + 0.2 + 0.6}{3} = \frac{0.9}{3} = 0.3
✅ Final Answer: 0.3
Explanation:
The restoration of road networks after an earthquake is critical for emergency response, supply delivery, and rebuilding efforts. Understanding the likelihood of different debris levels on roads helps planners allocate resources effectively. In the referenced study, debris conditions—categorized as light, moderate, or heavy—depend on the magnitude of the earthquake (low, moderate, or high), and these magnitudes are equally probable.
To determine the overall probability that a road is covered with heavy debris, we apply the Law of Total Probability. This law allows us to combine the conditional probabilities of heavy debris under each disaster magnitude with the likelihood of each magnitude occurring.
In this case, the conditional probabilities are:
- 0.1 for low magnitude,
- 0.2 for moderate magnitude,
- 0.6 for high magnitude.
Since each magnitude occurs with a probability of 1/3, the calculation is straightforward: P(Heavy)=(0.1)(1/3)+(0.2)(1/3)+(0.6)(1/3)=0.3P(\text{Heavy}) = (0.1)(1/3) + (0.2)(1/3) + (0.6)(1/3) = 0.3
This means that, on average, 30% of roads are expected to be covered with heavy debris following an earthquake. This figure is vital for emergency management agencies because roads under heavy debris often require the most time and equipment to clear. Knowing this probability enables better resource planning, such as pre-positioning heavy-duty clearing equipment or prioritizing key routes for immediate response. In summary, a clear understanding of probabilistic outcomes improves strategic disaster recovery and reduces downtime in critical infrastructure.
