Problem 1: Return Period and Exceedance Probability A river gauge has recorded 50 years of peak annual flow data. The largest flood on record was $1,200 \text{ m}^3/\text{s}$, which occurred exactly once. Calculate the empirical exceedance probability and the estimated return period of a $1,200 \text{ m}^3/\text{s}$ flood using the Weibull plotting position formula.

Problem 2: Risk and Reliability in Design A temporary cofferdam is being built to protect a bridge construction site. The construction project will take 3 years to complete ($n=3$). The engineer wants the risk of the cofferdam being overtopped during the construction period to be exactly 10% ($R = 0.10$). Determine the required design return period ($T$) for the flood that the cofferdam must withstand.

Problem 3: Gumbel's Extreme Value Distribution A river's annual maximum flood series has a mean ($\bar{x}$) of $4,500 \text{ m}^3/\text{s}$ and a standard deviation ($\sigma$) of $1,200 \text{ m}^3/\text{s}$. The sample size is $N = 60$ years. From statistical tables for $N=60$, the reduced mean ($\bar{y}_n$) is 0.5521 and the reduced standard deviation ($S_n$) is 1.1747. Estimate the magnitude of the 100-year flood ($x_{100}$) using Gumbel's method.

Problem 4: Log-Pearson Type III Distribution The logarithms (base 10) of an annual maximum flow series have been calculated. The mean of the log values ($\overline{\log x}$) is 3.50, the standard deviation ($\sigma_{\log x}$) is 0.25, and the skewness coefficient ($C_s$) is 0.4. Using a statistical table, the frequency factor ($K_z$) for a 50-year return period with a skew of 0.4 is 2.261. Calculate the magnitude of the 50-year flood.

Case Study 1: The "100-Year Flood" Misconception A homeowner buys a house in a designated "100-year floodplain." Five years later, the house floods. Three years after that, it floods again. The furious homeowner sues the city, claiming their hydrologists were wrong because a 100-year flood shouldn't happen twice in a decade. Explain the statistical reality underlying this situation.

Case Study 2: Stationarity vs. Climate Change in Frequency Analysis For decades, civil engineers designed dams, bridges, and levees based on historical gauge data, assuming the statistical properties of extreme floods remained constant over time. Discuss why climate change invalidates this core assumption and forces a paradigm shift in statistical hydrology.