  We consider the problem of counting the number of varieties in a family over
$\mathbb{Q}$ with a rational point. We obtain lower bounds for this counting
problem for some families over $\mathbb{P}^1$, even if the Hasse principle
fails. We also obtain sharp results for some multinorm equations and for
specialisations of certain Brauer group elements on higher dimensional
projective spaces, where we answer some cases of a question of Serre. Our
techniques come from arithmetic geometry and additive combinatorics.
