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
@Happy Mittal Can we also do this like this: First we calculate the expected value of shorter stick ( integration of x time 1 with limits 0 to 1. We now use this expected value as limits to a new expectation, which will the expected value of the shorter stick. I got the same answer using that. Just wanted to confirm.
Congratulations
Thank you