  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 $_3F_2$ hypergeometric functions. The O'Hara interesting
interpretation of this Clebsch-Gordan coefficient by binomial random variables
can also be related to hypergeometric functions ($_2F_1$), 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.
