  In this paper, we establish a strategy for the calculation of the proportion
of everywhere locally soluble diagonal hypersurfaces of $\mathbb{P}^{n}$ of
fixed degree. Our strategy is based on the product formula established by
Bright, Browning and Loughran. Their formula reduces the problem into the
calculation of the proportions of $\mathbb{Q}_{v}$-soluble diagonal
hypersurfaces for all places $v$. As worked examples, we carry out our strategy
in the cases of quadratic and cubic hypersurfaces. As a consequence, we prove
that around $99.99\%$ of diagonal cubic $4$-folds have $\mathbb{Q}$-rational
points under a hypothesis on the Brauer-Manin obstruction.
