This article introduces a new three-parameter discrete probability distribution, namely, the discrete hyperbolic mixture (DHM) distribution. It is derived by taking a mixture of two discrete distributions, called the hyperbolic cosine (HC) and hyperbolic sine (HS) distributions, which are also introduced here. Several probability and statistical properties of the proposed distributions are derived. Maximum likelihood estimation is employed to find estimators of the DHM distribution’s parameters. Numerical experiments are then conducted to evaluate the performance of the proposed estimator. The results show that the average estimate for each parameter approaches its true value as the sample size increases. The final section of this article presents applications of the DHM distribution to real datasets and performs a comparative study with some existing distributions to demonstrate its potential as an alternative model for count data. © 2025, Jomard Publishing. All rights reserved.