I'm going to assume that the distributions of the two points the segment is broken at areA line segment of length 5 is broken at two random points along its length. What is the probability that the shortest of the three new segments has length longer than 1?

a) independent

b) uniform[0,5]

Plotting this we see the following boolean contour plot, showing as a function of the break points $x,y$

whether the shortest segment has length less than 1.

those tannish triangles are the where the criterion fails to be met.

They have area $\dfrac{2\cdot 2}{2} = 2$ and there are two of them for a total area of $4$

The area of the probability space is $5\cdot 5=25$

so the probability that the criteria is met is

$P[\text{shortest segment has length <1}]=1 - \dfrac{4}{25} = \dfrac{21}{25}$