Answer: 48

Solution:

Each inscribed angle of the triangle subtends an arc of twice its measure, so the vertices split the circle into arcs of $2x,$ $2x,$ and $360 - 4x$ degrees. The chord fails to intersect the triangle exactly when both random points fall in the same arc, which has probability $$\begin{aligned} &\left(\frac{2x}{360}\right)^2 + \left(\frac{2x}{360}\right)^2 \\ &\quad {}+ \left(\frac{360 - 4x}{360}\right)^2 \\ &= 1 - \frac{14}{25} = \frac{11}{25}. \end{aligned}$$

Setting $y = \frac{x}{180},$ this reads $2y^2 + (1 - 2y)^2 = \frac{11}{25},$ which simplifies to $75y^2 - 50y + 7 = 0,$ with roots $y = \frac{1}{5}$ and $y = \frac{7}{15}.$ These give $x = 36$ and $x = 84,$ both legitimate base angles of an isosceles triangle.

The requested difference is $84 - 36 = 48.$