Let $(A_i,\tau_i)_{i\in I}$ be noncommutative probability spaces, let $A=*_{i\in I}A_i$ be their algebraic free product with amalgamated unit, and let $\tau=*_{i\in I}\tau_i$ be the free product state. For $a\in A_i$, write \begin{align*} a=\tau_i(a)1+a^\circ \end{align*} with $a^\circ\in A_i^\circ:=\ker(\tau_i)$. For every word $x_1\cdots x_n$ with $x_k\in A_{i_k}$, there exist $\lambda\in\mathbb C$ and finitely many reduced words $w_1,\dots,w_N$ such that \begin{align*} x_1\cdots x_n=\lambda 1+\sum_{r=1}^N w_r, \end{align*} and for every expression of this form, \begin{align*} \tau(x_1\cdots x_n)=\lambda. \end{align*} In particular, if $x_1\cdots x_n$ is already a reduced word, then $\tau(x_1\cdots x_n)=0$.