  A new approach to probability theory based on quantum mechanical and Lie
algebraic ideas is proposed and developed. The underlying fact is the
observation that the coherent states of the Heisenberg-Weyl, $su(2)$,
$su(r+1)$, $su(1,1)$ and $su(r,1)$ algebras in certain symmetric (bosonic)
representations give the ``probability amplitudes'' (or the ``square roots'')
of the well-known Poisson, binomial, multinomial, negative binomial and
negative multinomial distributions in probability theory. New probability
distributions are derived based on coherent states of the classical algebras
$B_r$, $C_r$ and $D_r$ in symmetric representations. These new probability
distributions are simple generalisation of the multinomial distributions with
some added new features reflecting the quantum and Lie algebraic construction.
As byproducts, simple proofs and interpretation of addition theorems of Hermite
polynomials are obtained from the `coordinate' representation of the (negative)
multinomial states. In other words, these addition theorems are higher rank
counterparts of the well-known generating function of Hermite polynomials,
which is essentially the `coordinate' representation of the ordinary
(Heisenberg-Weyl) coherent state.
