Yes, here's a nice and beautiful argument!
First , the way you define should draw a picture of axes
b. You're asked to select uniformly a point in the probability, square
[0,1]x[0,1]. Now because of the symmetry (sic!) it's equivalent to first choosing first point the points
a uniformly and then second point
b uniformly on [a, 1].
Now you should draw a picture of axes
a and in the triangle cut from the square by
b > a.You're
So you're actually uniformly selecting a point inside triangle defined by lines
Now let's find the conditions to be able to make a triangle of short sticks. We should have
a + (1-b) > b-a,
b-a + (1-b) > a and
b > 1 - b which indeed, as you say, boils down to
b > 1/2, a < 1/2, b-a < 1/2
It remains to note that those lines create inside the big triangle a small triangle which is similar to big but with all lengths
1/2 of the big, so this small triangle has area of exactly 1/4 of original!