I was doing some review on probability and came across the following exercise:
A quadratic equation $ax^2+bx+c=0$ is copied by a typist. However, the numbers standing for a, b and c are blurred and she can only see that they are integers of one digit. What is the probability that the equation she types has real roots?
Quadratic equations have real roots when the determinant is greater than or equal to $0$. Therefore $P(\text{real roots})=P(b^2-4ac\ge0)=P(b^2\ge4ac)$ My original thoughts were to find $1-P\left(\left(\frac{b}{2}\right)^2<4ac\right)$ where I would break it into cases where $b$ ranges from $0$ to $9$. I would then find out how many cases $4ac$ was larger than $b^2$ and divide it by $19^2*18$ (since $a\ne0$). When I found out it would be too tedious I thought of a possible geometric interpretation. Maybe the the probability could be expressed as ratios between volumes or something similar. When that seemed to over-complicate the problem I figured I was probably approaching it incorrectly.
Any hints or nudges in the general direction would be greatly appreciated.