  Consider tossing a collection of coins, each fair or biased towards heads,
and take the distribution of the total number of heads that result. It is
natural to conjecture that this distribution should be 'more random' when each
coin is fairer. Indeed, Shepp and Olkin conjectured that the Shannon entropy of
this distribution is monotonically increasing in this case. We resolve this
conjecture, by proving that this intuition is correct. Our proof uses a
construction which was previously developed by the authors to prove a related
conjecture of Shepp and Olkin concerning concavity of entropy. We discuss
whether this result can be generalized to $q$-R\'{e}nyi and $q$-Tsallis
entropies, for a range of values of $q$.
