Because question says so
at a point chosen uniformly at random
In GATE all information and hints will always be there in question. But many aspirants are used to solving substandard questions prepared in test series and hence skip those vital information.
For better intuition ..:)
What if the stick is broken randomly at two points? My approach:
To be shortest: length < 1/3
Pdf = 3 (using 1/b-a)
$3 \int_{0}^{1/3} L dx = 1/6$
Is 1/6 correct? Below link says it is 1/9:
https://math.stackexchange.com/questions/13959/if-a-1-meter-rope-is-cut-at-two-uniformly-randomly-chosen-points-what-is-the-av
Gatecse