Yes, sorry for the omission: I will edit. However, I am particularly interested, more generally, in counting the algebraic points of degree $d$ and height $< 1/d$. As for those of degree $d$ and height $< 1$, this is perhaps a much more delicate problem; I wonder if there are any non-trivial bounds available.

One more reference: this question, and its resolution in the resemblance to a small "r" (for "radix"), is also brought up in Barry Mazur's Imagining numbers (particularly the square root of minus fifteen), where further historical references may be found.

Here is one reference: the paper by Poonen and Rains, "Random maximal isotropic subspaces and Selmer groups." By the way, a trivial remark: you also get the same expectation $d/2$ within the very special family of products of elliptic curves. I wonder which of these two biases is closer to the truth.

Indeed... Sorry about this. I have edited again.

@ Felipe: Thanks! Let me ask you the question which came up in the above link. Given an appropriate number field $K$ (e.g. a polyquadratic field), would it indeed be true that the $K$-rank of a random elliptic curve over $\mathbb{Q}$ is strictly $> 1/2$?

It is better, in view of both comments, to say that $B$ is a quasi-projective variety over $\mathbb{Q}$, rather than a finite-type scheme over $\mathbb{Z}$. I will edit to adjust this.

It means the following. Let $h$ be a height function on $B$ given by some projective embeding (assume, to fix ideas, that $B$ is quasi-projective). Then the proportion is the lim, or the lim sup, as $H \to \infty$ of the ratio $|b \in Z \mid h(b) \leq H| / |b \in B(\mathbb{Z}) \mid h(b) \leq H|$, where $Z \subset B(\mathbb{Z})$ is the set of $b$ for which $X_b$ does have a rational point.

Edited: what I really wanted to say is that $X_{\mathbb{Q}} \to B_{\mathbb{Q}}$ was a smooth proper family of curves. So I have replaced "smooth" by "generically smooth."