We report an adjustable unbalanced quantum random-number generator based on the polarization of photons, which can produce nondeterministic true random unbalanced numbers. The underlying physical process is inherently quantum mechanical. To prove the quality of the output sequence of the proposed generator, we test the obtained bias-free sequence through the 3-standard-deviation criteria and the National Institutes of Standards and Technology test suite. Another type of nondeterministic unbalanced random-number generator is also studied in this work, to evaluate the quality of the output biased random numbers.