River stage forecasts are useful for decision makers to minimize flood damage. However, it is difficult to interpret the forecasts properly because of the forecast uncertainty. One of the biggest causes of the forecast errors is precipitation forecast errors. Here we propose a probabilistic distribution to model the precipitation forecast errors. The proposed distribution does not underestimate heavy precipitation and considers correlations among forecast horizons unlike existing studies. We also propose methods to evaluate confidence intervals of the river stage forecasts and excess probabilities of certain water levels using the proposed distribution. We applied the methods to Yoshino river system during the 2018 Japan floods. The methods yielded better Ignorance scores of excess probabilities and valid confidence intervals. The proposed methods can be applied to any river stage forecasting methods and are expected to be introduced to actual rivers for better decision making.