  We introduce and study a certain discrete probability distribution
$P_{n,m,k,l}$ having non-negative integer parameters $n,m,k,l$, motivated by
the asymmetry problem in quantum information theory. Its analysis reveals the
number of orthogonal vectors among permuted vectors of the tensor product of
asymmetric and invariant vectors. The distribution is defined by irreducible
decomposition of the tensor product $\Xi_{n,m|k,l}$ of a certain asymmetric
state and the Dicke state in the
$\operatorname{SU}(2)$-$\mathfrak{S}_n$-bimodule $(\mathbb{C}^2)^{\otimes n}$
appearing in the classical Schur-Weyl duality. We derive several explicit
formulas of the probability mass function $p(x)=p(x|n,m,k,l)$ involving
hypergeometric orthogonal polynomials via representation theoretic methods.
Among them, Racah presentation is the most remarkable and useful formula, which
expresses the pmf $p(x)$ by a single Racah polynomial. We also derive a formula
expressing the cumulative distribution function in terms of a terminating ${}_4
F_3$-hypergeometric series. Using these explicit formulas, we study asymptotic
behavior of $P_{n,m,k,l}$ in two types of limit $n \to \infty$. In the first
limit, we fix $k$, $l$ and $m/n$, and show that the asymptotic discrete
distribution is the convolution of two binomial distributions. In the second
limit, we fix $m/n$, $k/n$ and $l/n$, and prove the central limit theorem of
$P_{n,m,k,l}$, of which the limit normal distribution is derived by the
three-term recurrence relation of $p(x)$ coming from that of Racah polynomial.
We also describe the asymptotic behavior of the expectation and the variance
beyond the central limit theorem. Based on these asymptotic analysis, we
discuss the degree of asymmetry of the tensor product state $\Xi_{n,m|k,l}$. In
the appendix, we discuss $\mathbf{q}$-analogue of the computations on
$P_{n,m,k,l}$.
