$P(X^2+Y^2<1)$ when $X,Y\sim U[0,1]$

Question

Given that $X$ and $Y$ are iid $U[0,1]$, find $P(X^2+Y^2<1)$.

I've tried solving it by evaluating the following integral: $$\int_0^1\int_0^{\sqrt{1-y^2}}\,dx\,dy$$ Is this the correct way to solve it?

Answer 1

Integration is one correct way. For this question, though, it can be translated into geometric terms, which I find easier to visualise.

Draw the state space of $X$ and $Y$, which is a square, then mark out the region where $X^2+Y^2<1$, which is a quarter circle. The required probability is then the ratio of the quarter circle to the square, or $\frac\pi4$.

Answer 2

That is a correct way to solve it.

A better correct way to solve it, since you're dealing with uniform random variables on the unit square, is to ask "what is the area of the shape defined by $X^2+Y^2\leq 1$?"

Hint: the shape is a quarter circle.