A probability evolution method is proposed to quantify time-variant systems with mixed uncertainty based on a probability box. The cumulative distribution function(CDF) evolution is obtained from time-variant system response. The double-loop sampling method is used to separate for the epistemic uncertainties from the sampling of the aleatory uncertainties. The outer loop is for sampling of the epistemic uncertainties by Monte Carlo, and the inner loop is for sampling the aleatory uncertainties by a point-collocation non-intrusive polynomial chaos method. A time-variant probability box for system response can be obtained by the CDF boundary calculating at different time. The proposed method is verified through a delay performance degradation circuit. The studies demonstrate that the time-variant probability box not only quantifies the mixed uncertainty at each time, but also reflects the system response and uncertainty changing with time.