Question

The return period of a large earthquake for a given region is 200 years. Assuming that earthquake occurrence follows Poisson's distribution, the probability that it will be exceeded at least once in 50 years is ______ \% (rounded off to the nearest integer).

Hint:

The Poisson distribution is particularly useful for modeling the number of events in fixed intervals of time or space when these events occur with a known constant mean rate and independently of the time since the last event.

Answer 1

Step 1: Determine the average number of occurrences in 50 years. Since the return period of a large earthquake is 200 years, the average number of earthquakes in one year (\(\lambda\)) is: \[ \lambda = \frac{1}{200} \text{ earthquakes per year} \] Step 2: Calculate \(\lambda\) for 50 years. \[ \lambda_{50} = 50 \times \frac{1}{200} = 0.25 \] This means the expected number of large earthquakes in 50 years is 0.25. Step 3: Use the Poisson distribution formula to find the probability of having at least one earthquake. The probability of having exactly \(k\) earthquakes is given by: \[ P(X = k) = e^{-\lambda_{50}} \frac{\lambda_{50}^k}{k!} \] We need the probability of having at least one earthquake (\(P(X \geq 1)\)), so we calculate: \[ P(X \geq 1) = 1 - P(X = 0) \] \[ P(X = 0) = e^{-0.25} \frac{0.25^0}{0!} = e^{-0.25} \] Using a calculator for \(e^{-0.25}\): \[ P(X = 0) \approx 0.7788 \] So, \[ P(X \geq 1) = 1 - 0.7788 \approx 0.2212 \] Step 4: Convert the probability to a percentage and round to the nearest integer. \[ \text{Probability} \approx 22.12\% \] Rounded to the nearest integer, the probability is approximately 22\%. % Topic - Poisson distribution