Final answer:
The mean rate of bridge failure is 0.000668 per year, and the failure probability of the bridge in 70 years is approximately 4.55%, calculated using the Poisson distribution.
Step-by-step explanation:
Mean Rate of Failure and Failure Probability of a Bridge
To determine the mean rate of failure of the bridge, we need to consider both the rate of earthquake occurrence (λ = 0.01 per year) and the probability of the bridge failing when an earthquake occurs. Since the Factor of Safety (FOS) is normally distributed with a mean of 1.3 and a standard deviation of 0.20, the failure of the bridge corresponds to FOS values below 1. Using the standard normal distribution, we first find the Z-score corresponding to FOS = 1.
Z = (FOS - mean) / standard deviation = (1 - 1.3) / 0.20 = -1.5
The probability of FOS < 1 can be found from the standard normal distribution table, which gives the probability of about 0.0668 (or 6.68%). This is the probability that the bridge will fail given that there is an earthquake. To find the overall rate of bridge failure, we multiply this probability by the rate of earthquake occurrence:
Mean rate of bridge failure = earthquake rate * probability of bridge failure during an earthquake = 0.01 * 0.0668 = 0.000668 (or 0.0668% per year).
Next, to determine the failure probability of the bridge in 70 years, we can treat this as a Poisson distribution problem since we are dealing with rare events. The mean number of failures (λT) over 70 years is the rate of bridge failure times 70 years:
λT = mean rate of bridge failure * time period = 0.000668 * 70 = 0.04676
The probability of at least one failure occurring in 70 years is given by 1 minus the probability of zero failures, which is e^(-λT):
P(at least one failure in 70 years) = 1 - e^(-λT) = 1 - e^(-0.04676) ≈ 4.55%.
The mean rate of failure of the bridge is 0.000668 per year, and the failure probability of the bridge in 70 years is approximately 4.55%.