Let $AB$ be the line segment in question of length $l$ . Let $C$ and $D$ be the points which divide $AB$ into $3$ parts .

Let $AC=x,CD=y$ , Then $DB=l-x-y$

Clearly $x+y<l$ , therefore the sample space is given by the region enclosed by $\Delta OPQ$ , where $OP=OQ=l$ .

Area of $\Delta OPQ=\frac{l^{2}}{2}$

Now if the parts $AC,CD$ and $DB$ form a triangle , then $x+y>l-x-y \quad i.e. \quad x+y>\frac{l}{2} \quad (1)$ $x+l-x-y>y \quad i.e. \quad y<\frac{l}{2} \quad (2)$ $y+l-x-y>x \quad i.e. \quad x<\frac{l}{2} \quad (3)$

From $(1),(2)$ and $(3)$ , we get the event needed as the region enclosed by $\Delta RST$

Therefore the probability of the event is $\dfrac{ar( \Delta RST)}{ar( \Delta OPQ)} = \frac{\frac{1}{2} \frac{l}{2} \frac{l}{2}}{\frac{l^{2}}{2}} = \frac{1}{4}$

Well well ,now what was your guess for the Lucky Boy ?

Upvote the comment containing the name of your guess .

Well, time for some random Monte Carlo simulations in Python.

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from random import random
def triangle(a,b,c):
    sides = set([a,b,c])
    hypotenuse = max(sides)
    smaller = sides-set([hypotenuse])
    return sum(smaller) > hypotenuse
yes = 0.0
trials = 1000000
for trial in xrange(trials):
    slice_1 = random()
    slice_2 = random()
    if slice_1 > slice_2:
        slice_1,slice_2 = slice_2,slice_1
    a = slice_1
    b = slice_2 - slice_1
    c = 1 - slice_2
    if triangle(a,b,c):
        yes += 1
print "Answer:", yes/trials