When crossing an organism that is homozygous recessive for a single trait with a heterozygote, what is the chance of producing an offspring with the homozygous recessive phenotype?

Use a Punnett Square, heterozygous genotype we can use Tt and homozygous recessive we can  use  tt (T as a dominant trait and t as a recessive) thus follows:
  T  t
t Tt  tt
t Tt  tt

50% Tt and 50% tt, thus your answer is 50% chance of having an offspring with the homozygous recessive phenotype in this case (tt).