Final answer:
The risk of at least one 100-year flood occurring during the 20-year lifespan of the culvert is the complement of the probability that the flood does not occur over the 20 years. This is calculated as 1 - (0.99)^20, which means option A is the correct answer. Option a is correct.
Step-by-step explanation:
The risk associated with the hydrologic design of a culvert that is expected to handle a 100-year flood but has a life span of 20 years is calculated by considering the probability that such a flood will occur at least once in the 20-year period. The probability of a 100-year flood happening in any given year is 1/100, or 0.01.
The probability of the flood not happening in a single year is therefore 1 - 0.01 = 0.99. We must calculate the probability that the flood does not occur in any of the 20 years and subtract that from 1 to find the overall risk (probability) of at least one such flood occurring over the 20-year lifespan of the culvert.
To calculate this, we take the probability of no flood in a single year (0.99) and raise it to the power of 20 to find the probability of no flood over 20 years. The risk of at least one flood occurring over that period is then 1 minus this probability:
Risk = 1 - (0.99)^20
The correct answer to the student's question is Option A: 1 - (0.99)^20.