A particular case of degenerate Clebsch-Gordan coefficient can be expressed with three binomial coefficients. Such a formula, which may be obtained using the standard ladder operator procedure, can also be derived from the Racah-Shimpuku formula or from expressions of Clebsch-Gordan coefficients in terms of
3F2 hypergeometric functions. The O'Hara interesting interpretation of this Clebsch-Gordan coefficient by binomial random variables can also be related to hypergeometric functions (
2F1), in the case where one of the parameters tends to infinity. This emphasizes the links between Clebsch-Gordan coefficients, hypergeometric functions and, what has been less exploited until now, the notion of probability within the framework of the quantum theory of angular momentum.