How to prove that the probability that a randomly selected point in a square lies inside the circle inscribed in the square is equal to the ratio of the area of the circle and square ?
To rephrase the question in a better sense, suppose $X$ and $Y$ be the independent random variables representing the $x$ and $y$ co-ordinates of the point. Here $X$ and $Y$ are uniformly distributed between $[-a,a]$ where the square has vertices with co-ordinates $(\pm a, \pm a)$. Now I want to find the probability $P(X^{2}+Y^{2} < a^{2})$. Hope I have made my question clear.