= Solution
Parity conservation and the negative parity of $X$ require odd <orbital angular momentum> $L$. Since the two $Y$ particles are identical bosons, their spin state must then be antisymmetric, so their total spin is $S=1$. Total angular momentum zero forces $L=S=1$.
Conditioning on spin projection $m_S=1$ leaves orbital projection $m_L=-1$. The angular density is therefore proportional to $|Y_{1,-1}(θ,φ)|^2\propto\sin^2θ$. Including the solid-angle measure, the requested probability is
$$
{\int_{π/4}^{3π/4}\sin^3θ\,dθ\over\int_0^π\sin^3θ\,dθ}
= {5\sqrt2\over8}.
$$
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