  In this paper we study the mixing time of a biased transpositions shuffle on
a set of $N$ cards with $N/2$ cards of two types. For a parameter $0<a \le 1$,
one type of card is chosen to transpose with a bias of $\frac{a}{N}$ and the
other type is chosen with probability $\frac{2-a}{N}$. We show that there is
cutoff for the mixing time of the chain at time $\frac{1}{2a} N \log N$. Our
proof uses a modified marking scheme motivated by Matthews' proof of a strong
uniform time for the unbiased shuffle.
