On a generalization of derangement polynomials and numbers

In T. Kim, D. S. Kim, and D. V. Dolgy, Probabilistic derangement numbers and polynomials , Math. Comput. Model. Dyn. Syst. 31 (2025), no. 1, 2529188, Kim-Kim defined the probabilistic derangement polynomials and numbers and found some properties of those polynomials and numbers. In this paper, we define another probabilistic derangement polynomials which are different from the one defined by Kim-Kim, and find explicit formulas and interesting identities of Stirling numbers of the first and the second kind, Bell polynomials, Fubini polynomials, Stirling numbers of the first and the second kind and those polynomials and numbers.