Solution:

By symmetry it will suffice to consider one quarter of the dartboard, which is a square of side length $1$. Therefore the probability is the area of the desired region in this square. The desired region is the part of the circle of radius $1$ centered at a corner that is closer to the opposite corner. The points closer to the opposite corner are those that are on the other side of the diagonal through the other two corners, so the desired region is a quarter of a circle of radius $1$ minus a right triangle with legs of length $1$. Therefore the area (and hence the probability) is $\frac{\pi-2}{4}$.