The Q-Coder is an important new development in binary arithmetic coding. It combines a simple but efficient arithmetic approximation for the multiply operation, a new formalism which yields optimally efficient hardware and software implementations, and a new technique for estimating symbol probabilities which matches the performance of any method known. This paper describes the probability-estimation technique. The probability changes are estimated solely from renormalizations in the coding process and require no additional counters. The estimation process can be implemented as a finite-state machine, and is simple enough to allow precise theoretical modeling of single-context coding. Approximate models have been developed for a more complex multi-rate version of the estimator and for mixed-context coding. Experimental studies verifying the modeling and showing the performance achieved for a variety of image-coding models are presented.