Two numbers are randomly selected between $(0,1)$, uniformly and independently distributed. What is the probability that the three resulting line segments, which are obtained by cutting the interval at the two selected numbers, form a triangle?
So, I was thinking I can do something like this:
$X \sim U(0,1), Y \sim U(0,1),$ which are independent. In order to form a triangle, each side cannot be bigger than the sum of the other two sides. So, each segment cannot be greater than $1/2$.
If $X > Y$, the first segment is of length $Y$, the second is of length $X-Y$, and the third is of length $1-X$.
So, can we obtain the answer by computing $2P(Y,X-Y,1-x < 1/2)$? If so, how would you compute this probability?