Let $P$ be the probability that an elliptic curve with a rational point has an infinite number of rational points. From what I understand, the value of P is unknown.

This got me thinking about a similar question related to Galois theory. Let $N$ be the probability that a finite extension of $\mathbb{Q}$ is a normal extension. Is anything known about the value of $N$?.

I wouldn't expect it to be 1 or 0.

My gut feeling is that not much is known as it seems related to the inverse Galois problem. Indeed, if every finite group were the galois group of some finite galois extension of $\mathbb{Q}$ i would imagine that $R$ would be somehow related to the "probability of a subgroup being normal" (this is intentionally vague).

Question: Is anything (if anything) known about $N$?