  A randomisation of the Berele insertion algorithm is proposed, where the
insertion of a letter to a symplectic Young tableau leads to a distribution
over the set of symplectic Young tableaux. Berele's algorithm provides a
bijection between words from an alphabet and a symplectic Young tableau along
with a recording oscillating tableau. The randomised version of the algorithm
is achieved by introducing a parameter $0 < q < 1$. The classic Berele
algorithm corresponds to letting the parameter $q \to 0$. The new version
provides a probabilistic framework that allows to prove Littlewood-type
identities for a $q$-deformation of the symplectic Schur functions. These
functions correspond to multilevel extensions of the continuous $q$-Hermite
polynomials. Finally, we show that when both the original and the $q$-modified
insertion algorithms are applied to a random word then the shape of the
symplectic Young tableau evolves as a Markov chain on the set of partitions.
