probability that a randomly selected point in a square lies inside the circle 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 $P(X^{2}+Y^{2} &lt; a^{2})$. Hope I have made my question clear.
Well although I doesn't make sense to put the words prove and probability in the same sentence, I'll answer this anyway. The question should be rephrased as find the probability and not prove the probability. 

Anyway, the simplest definition of probability is No. of favorable outcomes divided by Total NO. of outcomes. Since, we are selecting a point from the square, the area of the square is the total no of outcomes. And, the favorable is the area of the circle, since, that is where we want the point. So, answer is area of circle divided by area of square.
Well although I doesn't make sense to put the words prove and probability in the same sentence, I'll answer this anyway. The question should be rephrased as find the probability and not prove the probability. 

Anyway, the simplest definition of probability is No. of favorable outcomes divided by Total NO. of outcomes. Since, we are selecting a point from the square, the area of the square is the total no of outcomes. And, the favorable is the area of the circle, since, that is where we want the point. So, answer is area of circle divided by area of square.