<p>In this paper, we introduce a new mixed binomial distribution obtained by assuming that the success probability P in a binomial model follows the Log-Lindley distribution. The resulting model, which we refer to as the Binomial Log-Lindley (BLL) distribution, is governed by three interpretable parameters and offers enhanced flexibility over the classical binomial distribution. This added flexibility allows the model to accommodate a wider range of dispersion patterns, including overdispersion and zero inflation, making it a competitive alternative in discrete data modeling. We derive key statistical properties of the BLL distribution, including its probability mass function, cumulative distribution function, moments, and hazard-related functions. Parameter estimation is addressed via both the method of moments and maximum likelihood estimation, with special attention given to the case of censored data a common scenario in reliability and survival analysis. A comprehensive simulation study is conducted to assess the finite-sample performance and robustness of the proposed estimators. Finally, we illustrate the practical utility of the BLL model through applications to real-world datasets of traffic accident fatalities. Comparative analyses demonstrate that the BLL model provides a substantially improved fit over classical and other flexible discrete models, validating its potential as a valuable addition to the toolkit of discrete probability distributions.</p>