Shannon's concept of information is related to predictability. In a binary series, the value of information relies on the frequency of 0's and 1's, or how it is expected to occur. However, information entropy does not consider the bias in randomness related to autocorrelation. In fact, it is possible for a binary temporal series to carry both short- and long-term memories related to the sequential distribution of 0's and 1's. Although the Hurst exponent measures the range of autocorrelation, there is a lack of mathematical connection between information entropy and autocorrelation present in the series. To fill this important gap, we combined numerical simulations and an analytical approach to determine how information entropy changes according to the frequency of 0's and 1's and the Hurst exponent. Indeed, we were able to determine how predictability depends on both parameters. Our findings are certainly useful to several fields when binary times series are applied, such as neuroscience to econophysics.