A positive proportion of hyperelliptic curves have no unexpected quadratic points

We prove that when even-degree hyperelliptic curves are ordered by the sizes of their coefficients, a positive proportion of them have no unexpected quadratic points — i.e., no points defined over quadratic fields except for those that arise by pulling back rational points from P^1. To obtain this result, we combine a generalization of Selmer-group Chabauty (due to Poonen-Stoll) with new results on the average size of the 2-Selmer groups of Jacobians of even-degree hyperelliptic curves. This is joint work with Manjul Bhargava, Jef Laga, and Arul Shankar.