This paper proposes a flexible probability mass function for modeling count data, particularly over-dispersed and asymmetric observations. A novel two-parameter discrete distribution, the discrete power inverted Topp–Leone distribution, is presented using a survival discretization technique. The following statistical characteristics are examined: factorial moments, probability-generating function, quantiles, mean, variance, mean residual life, and entropy measures. The best estimators of the unknown parameters were obtained using various techniques, such as maximum likelihood, moments, least squares, Anderson–Darling, and Cramér-von Mises. A simulation study showed that the accuracy of the estimates improves with larger samples, although higher parameter values may affect precision. The findings indicate that the efficiency of these estimation methods varies under different conditions. Applications to liver lesions, chromatid aberrations, and criminal sociology datasets confirm the model's usefulness for discrete count data in fields such as social sciences, pharmacology, and environmental health. Finally, for modeling count data, the new probabilistic model can be employed as a competitive alternative to other existing distributions.