  $q$-analogs of special functions, including hypergeometric functions, play a
central role in mathematics and have numerous applications in physics. In the
theory of probability, $q$-analogs of various probability distributions have
been introduced over the years, including the binomial distribution. Here, I
propose a new refinement of the binomial distribution by way of the quantum
binomial theorem (also known as the the noncommutative $q$-binomial theorem),
where the $q$ is a formal variable in which information related to the sequence
of successes and failures in the underlying binomial experiment is encoded in
its exponent.
