University of GarmianPasser Journal of Basic and Applied Sciences270659442220200901Beta–Binomial and Overdispersion with Exchange of the Sample Size over the Probability Interval [0, 1] with Applications636811821410.24271/psr.13ENHanaw AhmedAminDepartment of Mathematics, College of Science, University of Sulaimani, Sulaimani, Kurdistan Region, Iraq0000-0002-4072-0294Rando RasulQadirDepartment of Mathematics, College of Basic Education, University Of Sulaimani, P.O. Box: 46, Sulaimani, Kurdistan Region, IraqJournal Article20200820The beta-binomial model that is generated by a simple mixture model has been commonly applied in the health, physical, and social sciences. In clinical and public health, overdispersion occurs due to biological variation between the subjects of interest. Both the binomial and beta-binomial models are applied to different problems occurring in rational test theory. In this study, we focused on modeling overdispersion for binomial distribution. The main aim was to show a complete and extensive understanding of the beta-binomial model and updated form by broaden its practical applications in the field of breast cancer with hormone medication. It is observed in different independent Bernoulli trials yes/no (<em>x<sub>i</sub></em>=1, 0) experiments with success probabilities 0<<em>p <sub>i</sub></em>< 1 and compare the model in a sequence of <em>n<sub>i</sub></em>. The performance of the maximum likelihood estimates technique that is used in moderate and small samples <em>n<sub>i</sub></em> by a Newton-Raphson iterative method using Matlab package. We have found that using hormones for other treatments have complication leading to breast cancer. We took 20 investigational testers in Hiwa Hospital for cancer treatment in Sulaymaniyah province, with proportion <em>p <sub>i</sub></em> is varying from 9.7% to 50 %. In addition, we concluded that the beta-binomial theory is a good alternative of binomial model. This is due to the fact that the beta-binomial model has provided a robust estimate for events from heterogeneous binomial studies. https://passer.garmian.edu.krd/article_118214_3ffe9073f2377a1f3f85973dd6e0193d.pdf