  Under two assumptions, we determine the distribution of the difference
between two functions each counting the numbers < x that are in a given
arithmetic progression modulo q and the product of two primes. The two
assumptions are (i) the Extended Riemann Hypothesis for Dirichlet L-functions
modulo q, and (ii) that the imaginary parts of the nontrivial zeros of these
L-functions are linearly independent over the rationals. Our results are
analogs of similar results proved for primes in arithmetic progressions by
Rubinstein and Sarnak. In particular, we show that the bias for products of two
primes is always reversed from the bias for primes. For example, while
Rubinstein and Sarnak showed, under (i) and (ii), that 99.6% of the time there
are more primes up to x which are 3 mod 4 than those which are 1 mod 4, we show
that 89.4% of the time, there are more products of two primes which are 1 mod 4
than 3 mod 4.
