The Okamoto tau function of the Painleve equations is based on a Hamiltonian for each of the six Painleve equations, and the corresponding tau function is defined by the requirement that the Hamiltonian is equal to the logarithmic derivative of the tau function. Certain integer sequences in parameter space are such that the tau functions are related by the toda lattice equation, and furthermore permit a trivial solution (tau[0]=1) and a classical solution (tau[1]=classical function). This allows the complete tau function sequence to be expressed as a determinant, which can be shown to be equal to a multiple integral representing an average in classical random matrix ensembles with unitary symmetry. New results obtained include the tau evaluation of the probability distribution (in contrast to the cumulative distribution known from earlier work) of the smallest eigenvalue in the LUE, and the largest eigenvalue in the LUE and GUE.