I have one doubt in the posted solutions. We have one more ways to calculate probability of each point in 2-D plane $\rightarrow$
For ex. P = $\frac{1}{2}$ will come via two points $\left ( \frac{1}{2}, 0 \right ) and \left (0, \frac{1}{2} \right )$ respectively. Now probability of obtaining each point is $\frac{1}{2}*1$ and $\frac{1}{2}*1$ (because X and Y both are following uniform distribution and both are independent). So total probability of getting P = $\frac{1}{2}$ is $\left ( \frac{1}{2} + \frac{1}{2} \right )$ . But via mentioned method probability for P = $\frac{1}{2}$ will be $\frac{1}{\sqrt{5}}$ .
Where am i doing mistake ?