A refinement of the binomial distribution using the quantum binomial theorem

$q$-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.