How to calculate the proportion of everywhere locally soluble diagonal hypersurfaces

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 $99.99\mathrm{\%}$ of diagonal cubic $4$-folds have $\mathbb{Q}$-rational points under a hypothesis on the Brauer-Manin obstruction.