Stable operation of a continuous casting process requires precise control of molten steel temperature in a tundish (TD temp), which is a container used to feed molten steel into an ingot mold. Since TD temp is implicitly controlled by adjusting molten steel temperature in the preceding secondary refining process (RH temp), a model relating TD temp with RH temp is required. This research proposes a procedure to predict the probability distribution of TD temp by integrating a gray-box model and a bootstrap filter to cope with uncertainties of the process. The derived probability distribution is used not only to predict TD temp but also to evaluate the reliability of prediction. The proposed method was validated through its application to real operation data at a steel work, and it was confirmed that the developed model satisfied the requirements for its industrial application.