The square of a single error poorly approximates error variance. If one could find numerous historical forecasts like today's forecast, the variance of the associated forecast errors would approximate the true forecast error variance given today's forecast. However, chaos makes it difficult to obtain relevant similar forecasts. In such situations the true error variance is not directly observable. It is hidden. Ensemble forecasts provide imperfect estimates of these hidden error variances. Here, we present apparently new analytic solutions for the distribution of true error variances given an imperfect error variance prediction. “Straw man” experiments are used to assess the value of knowledge of various aspects of this true distribution to ensemble post-processing. For example, is it enough to know the mean of this distribution? What is the value of information about its higher moments? The analytic solution pertains to a system which features fluctuating true forecast error variances that are stochastically linked to ensemble variances. Rank Frequency Histograms (RFH) and the effective daily interest rates associated with a hypothetical game of weather roulette are used to quantify the value of accurate information about the distribution of true error variances given the ensemble variance. It is found that ensemble post-processing schemes that utilize the full distribution of true error variances given the ensemble sample variance outperform those that do not. Specifically, when the ensemble accounts for the variance of true error variance, we obtain a flat RFH and an optimal effective daily interest rate.