Probably the set of $E/\mathbb{Q}$ with $E(\mathbb{Q})=\{O\}$ has density 1/2 (in an suitable way of ordering curves). A recent (and very deep) result of Manjul Bhargava and Arul Shankar proves that this set has positive density, so there are provably a lot of curves with the property that you request. There's a nice overview of the Bhargava-Shankar result, with some of the history, by Bjorn Poonen. It's freely available at
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