The Bell numbers arise in diverse combinatorial problems. Their study and those of the accompanying Bell polynomials are facilitated via their generating functions and results like the Dobinski formula. In this paper, the authors introduce and study generalizations of the above-mentioned numbers and polynomials with the aid of the hypergeometric function. For these generalizations, they obtain recurrence formulae, generating functions and integral expressions etc. These are used to prove some combinatorial identities as well. Interestingly, the authors also discuss probabilistic interpretations. For instance, they show that \(p\)-Bell polynomials \(B_{n,p}(x)\) can be viewed as the \(n\)-th moment of a random variable \(x + Z\) where \(Z\) follows the beta-Poisson law.