Answer : 1/2

By symmetry, we can reduce the problem to examining only the chord with a chosen direction.

Then, the factor which decides whether the chord is smaller or greater than the square root of 3 is the distance to the center of the circle.
The chord is greater than the square root of 3 if and only if its distance to the center of the circle is smaller than 1/2.

So, the probability is 1/2 .

Here is an animation that explains the way we can get to this answer:

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