There are two lines $y=m_1x+b_1$ and $y=m_2x+b_2$; their $y$-intercepts must be between $-1$ and 1, inclusive ($-1\le b_1 \le 1$ and $-1\le b_2 \le 1$). Thus, their intersection (given by $\left(\frac{b_{2}-b_{1}}{m_{1}-m_{2}},\ \frac{m_{1}b_{2}-m_{2}b_{1}}{m_{1}-m_{2}}\right)$) may fall inside, outside, or on the unit circle, defined by the equation $x^2+y^2=1$. I'm wondering if there is a non-calculus way to determine the probability that lines with uniformly randomly chosen values of $m_1$, $m_2$, $b_1$, and $b_2$ will end up intersecting inside the unit circle.
Thanks!