A number of writers have supposed that for the full specification of belief, higher order probabilities are required. Some have even supposed that there may be an unending sequence of higher order probabilities of probabilities of probabilities.... In the present paper we show that higher order probabilities can always be replaced by the marginal distributions of joint probability distributions. We consider both the case in which higher order probabilities are of the same sort as lower order probabilities and that in which higher order probabilities are distinct in character, as when lower order probabilities are construed as frequencies and higher order probabilities are construed as subjective degrees of belief. In neither case do higher order probabilities appear to offer any advantages, either conceptually or computationally.
My daughter just started a business analytics Master's program. For the probability sequence of the core statistics course, one of her assignments is to calculate the probability of single 5 card draw poker hands from a 52-card deck. I well remember this exercise from back in the day, when I computed all such relevant probabilities using basic combination counting techniques for an intro to probability course. My daughter though, a business undergrad, is less interested in the math than she is in the stats/computation, opining that's where she'll make money. It's hard to argue that logic, though I thought it might be an analytics "ah-hah" moment for her to connect the probability math with statistics.
de Finetti Bruno, (1992 ), "On the Subjective Meaning of Probability," in Paola Monari & Daniela Cocchi (eds), Bruno de Finetti: Probabilità e induzione (Induction and Probability), Bologna, CLUEB, 298-329. Title Link: Maria Carla Galavotti. "Pragmatism and the Birth of Subjective Probability". European Journal of Pragmatism and American Philosophy [Online], XI-1 | 2019, Online since 19 July 2019, connection on 21 July 2019.