First thing is to phrase the question in the notation of a probability. We are looking for -
p(x1<x2,x3<x2)
Next step is to simply expand this based on the rules of conditional probability as -
p(x1<x2,x3<x2)=â«10p(x1<x2,x3<x2|x2)p(x2)dx2
Given we are working with uniform[0,1] variables, the first term can be written simply -
p(x1<x2,x3<x2|x2)=x22
We also know that p(x2)=1 on the interval we care about.
Therefore we have -
p(x1<x2,x3<x2)=â«10x22dx2=x33âŁâŁâŁ10=1/3
