Probability of two uniform random numbers being more than $\frac{1}{2}$ apart

The condition that the two number chosen randomly from unit interval$(0, 1)$ is same as the conditions $0\lt x. y \lt 1$ . Let the two numbers be $x, y$ Then the probability is that $|x-y| \gt 0. 5$Draw the graph of the two functions and use geometric probability to get answer.

You need to just divide the sum of areas of two corner triangles with the square. It is easy to see probability is $0. 25$

The square mentioned in the problem is the sample space of the problems, where every point represents some outcome of picking the two random numbers. The blue regions in the plot below shows the regions where, for some $x$, $|y-x|>\frac{1}{2}$, which is bounded by the two functions in the problem formulation. The area of these regions gives the sought probability.

You have already had several answers but none with picture. A good practice is to try and visualize the problem with an image or graph or picture.

Here is a plot of the area you will want to integrate on your density function.