On some extremal problems for mutual information and entropy

The problem of determining the maximum mutual information $I(X;Y)$ and minimum entropy $H(X,Y)$ of a pair of discrete random variables $X$ and $Y$ is considered under the condition that the probability distribution of $X$ is fixed and the error probability $\mathrm{Pr}\{Y\ne X\}$ takes a given value $\varepsilon$, $0\le\varepsilon\le1$. Precise values for these quantities are found, which in several cases allows us to obtain explicit formulas for both the maximum information and minimum entropy in terms of the probability distribution of $X$ and the parameter $\varepsilon$.