In recent times, there has been significant attention to the improvements to classical probability distributions in the literature. Because of this attention, many probability generators or family distribution functions have emerged. These generators can control the skewness, kurtosis, and tail weight of datasets effectively. The unit modified Burr III family of distributions—an entirely new family of distributions—has been developed. The mathematical and statistical characteristics of this new family have been derived. Two special distributions, the unit modified Burr III Weibull (UMBIIIW) and unit modified Burr III Chen (UMBIIIC) distributions, have been developed from this family. These distributions’ parameters have been estimated using the maximum likelihood estimation (MLE) technique. A simulation method based on Monte-Carlo was employed to investigate the behavior of these estimators. Next, a real-life dataset was used to demonstrate the UMBIIIC distribution’s applicability. Finally, an entirely new regression equation, called the log unit modified Burr III Weibull (LUMBIIIW) regression equation, was proposed, and real-life data were used to demonstrate how applicable it is. With the data used in the application illustration, the new regression equation offers the best fit among its competitive models.