Random quantum processes with a countable set of elementary orthogonal events that govern the operation of a quantum computer are discussed. For the case of pure states, expressions for a posteriori quantum probabilities are derived. For stationary and nonstationary random quantum processes with the countable set of states, properties of transition quantum probabilities are investigated. A method for finding a posteriori quantum probabilities is proposed. An analog of the Kolmogorov equation for random quantum processes is presented.