We study a general tensor product for two collections of related physical operations or observations. This is a free product, subject only to the condition that the operations in the first collection fail to have any influence on the statistics of operations in the second collection and vice versa. In the finite-dimensional case, it is shown that the vector space generated by the probability weights on the general tensor product is the algebraic tensor product of the vector spaces generated by the probability weights on the components. The relationship between the general tensor product and the tensor product of Hilbert spaces is examined in the light of this result. We show how anomalous states arise (states that seem to assign ``negative probabilities''), how they can be interpreted, and how they can be dealt with mathematically. For totally finite systems, the main result is used to determine the dimensions of various other products that are related to the general tensor product. Alexander Wilce (Thesis, University of Massachusetts, 1988) has shown the more general result that (without any finiteness condition), the algebraic tensor product is always point-wise dense in the general tensor product.